This document provides code used to generate results and figures shown in the paper “Inferring time of infection from field data using dynamic models of antibody decay”.
For more information and details, please see the paper and the supporting information document.


1 Import and prepare data

observed.data.for.fitting = readRDS("Antibody_decay_data.RDS")
observed.data.for.fitting.incl.neg = readRDS("Antibody_decay_data_incl_neg.RDS")
neg.intervals = observed.data.for.fitting.incl.neg$time[which(observed.data.for.fitting.incl.neg$seroconversion.interval==T)]


kbl(observed.data.for.fitting) %>%
       kable_classic(bootstrap_options  =  c("striped", "hover","condensed"), full_width=F,html_font="Cambria",fixed_thead=T) %>%
       scroll_box(height="400px")
Pittag id time titer.pomona titer.aut lab
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00141 1 586 2 NA 1
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01334 5 0 6 NA 1
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01520 6 0 4 NA 1
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01619 7 352 3 NA 1
01619 7 699 3 NA 1
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89282 253 94 5 7 0
89282 253 363 4 6 0
89282 253 713 5 5 0
89282 253 1189 4 NA 1
89282 253 1469 3 NA 1
89282 253 1819 3 NA 1
89282 253 1935 3 NA 1
89282 253 2197 4 NA 1
90599 254 0 9 9 0
90599 254 351 7 9 0
90599 254 352 7 7 0
90599 254 382 7 7 0
90599 254 701 2 2 0
90599 254 1105 2 NA 1
90599 254 1434 1 NA 1
90697 255 0 8 8 0
90697 255 121 7 8 0
90697 255 406 7 NA 1
91036 256 0 5 NA 0
91036 256 371 4 NA 0
91036 256 1939 2 NA 1
91426 257 0 6 NA 1
91426 257 379 4 NA 1
92210 258 0 7 NA 0
92210 258 425 3 NA 0
92210 258 837 3 NA 0
92299 259 0 4 4 0
92299 259 379 3 3 0
92299 259 801 2 2 0
92299 259 1112 2 3 0
92299 259 1463 0 1 0
93678 260 0 8 9 0
93678 260 918 3 NA 0
93706 261 0 5 5 0
93706 261 664 4 NA 0
93706 261 708 4 NA 0
95733 262 0 6 NA 0
95733 262 362 6 NA 0
95733 262 1191 4 NA 1
96261 263 0 5 9 0
96261 263 308 4 4 0
96261 263 1418 2 NA 0
96353 264 0 4 NA 0
96353 264 482 4 NA 0
96353 264 816 3 NA 0
96382 265 0 5 5 0
96382 265 351 3 3 0
96382 265 756 3 NA 1
96382 265 1085 2 NA 1
96382 265 1435 0 NA 1
96382 265 1592 0 NA 1
96382 265 1818 1 NA 1
96419 266 0 2 NA 1
96419 266 349 1 NA 1
96B79 267 0 4 4 0
96B79 267 1242 2 NA 0
98152 268 0 7 7 0
98152 268 742 4 4 0
98195 269 0 6 NA 1
98195 269 349 2 NA 1
98521 270 0 3 NA 1
98521 270 345 2 NA 1
99121 271 0 5 NA 0
99121 271 370 1 NA 0
99121 271 869 2 NA 0
99842 272 0 8 10 0
99842 272 537 6 NA 0
A2B2C 273 0 4 5 0
A2B2C 273 159 4 4 0
A2B2C 273 368 2 3 0
A2B2C 273 3081 0 NA 1
A3039 274 0 4 7 0
A3039 274 1090 2 NA 0
A543D 275 0 8 9 0
A543D 275 444 4 NA 0
A543D 275 752 4 NA 0
A6234 276 0 6 8 0
A6234 276 356 4 5 0
A6234 276 1113 2 4 0
A6234 276 1242 2 4 0
A6234 276 1551 2 5 0
A6234 276 1847 3 4 0
A6234 276 2221 3 NA 1
A712D 277 0 6 5 0
A712D 277 312 3 3 0
B0712 278 0 7 8 0
B0712 278 1122 2 NA 0
B0712 278 1563 3 NA 0
B0712 278 1858 3 NA 0
B0F3B 279 0 5 7 0
B1311 280 0 4 2 0
B1311 280 331 2 1 0
B1311 280 1283 0 0 0
B1311 280 1660 0 0 0
B6069 281 0 9 9 0
B6069 281 661 2 3 0
C0102 282 0 4 6 0
C0102 282 323 5 5 0
C0102 282 676 4 6 0
C0102 282 1463 4 NA 0
C0D60 283 0 5 1 0
C0D60 283 5 5 5 0
C0D60 283 737 3 3 0
C0D60 283 866 2 3 0
C0D60 283 1087 2 2 0
C0D60 283 1478 2 2 0
C3530 284 0 5 6 0
C3530 284 671 4 4 0
C667D 285 0 7 7 0
C667D 285 163 6 7 0
C667D 285 220 7 7 0
C667D 285 369 6 7 0
C667D 285 757 7 8 0
C667D 285 1118 7 NA 0
D2B53 286 0 10 6 0
D2B53 286 987 1 4 0
D353A 287 0 9 9 0
D353A 287 347 6 6 0
D353A 287 942 2 4 0
D353A 287 1309 2 0 0
D353A 287 1663 1 2 0
D6642 288 0 4 3 0
D6642 288 343 3 2 0
D6642 288 386 3 2 0
D6642 288 747 0 0 0
D7A14 289 0 3 0 0
D7A14 289 1461 2 NA 1
E022A 290 0 4 NA 0
E022A 290 356 4 NA 0
E022A 290 722 3 NA 0
E0472 291 0 7 8 0
E0472 291 951 3 5 0
E0472 291 1330 3 4 0
E0472 291 1722 1 3 0
E0472 291 2065 2 NA 0
E0472 291 2414 3 NA 0
E1154 292 0 6 7 0
E1154 292 252 5 6 0
E1154 292 629 4 5 0
E1B15 293 0 2 4 0
E1B15 293 737 1 2 0
E1B15 293 1478 0 NA 0
E1B15 293 1851 0 NA 0
E1D7D 294 0 9 10 0
E1D7D 294 361 5 6 0
E1D7D 294 1014 2 4 0
E2204 295 0 5 7 0
E2204 295 321 3 NA 0
E2204 295 501 4 NA 0
E2204 295 859 4 NA 0
E6D47 296 0 5 NA 1
E7E64 297 0 8 8 0
E7E64 297 985 4 5 0
E7E64 297 1485 3 5 0
F1071 298 0 6 4 0
F1071 298 147 5 5 0
F1071 298 304 1 3 0
F1071 298 534 2 3 0
F1071 298 711 2 3 0
F1555 299 0 7 7 0
F1555 299 321 0 4 0
F5F25 300 0 7 7 0
F5F25 300 257 6 8 0
F5F25 300 422 6 7 0
F5F25 300 626 5 7 0
F5F25 300 1407 3 NA 0
03596 301 0 0 1 0
03596 301 347 0 1 0
03596 301 694 0 0 0
03596 301 1086 0 0 0
03596 301 2187 0 NA 1
13726 302 0 0 2 0
13726 302 786 0 2 0
13726 302 1486 0 1 0
13726 302 1933 0 NA 1
2184D 303 0 2 5 0
2184D 303 1307 0 3 0
41B20 304 0 1 2 0
41B20 304 343 0 1 0
41B20 304 683 0 3 0
41B20 304 1028 0 0 0
41B20 304 1813 0 NA 0
B1757 305 0 4 4 0
B1757 305 724 0 2 0
B1757 305 1101 0 NA 0
B1757 305 2659 0 NA 1
B1757 305 2925 0 NA 1
C6F06 306 0 3 4 0
C6F06 306 482 0 1 0
C6F06 306 1112 0 NA 0
C6F06 306 1825 0 0 0
D0701 307 0 0 1 0
D0701 307 917 3 3 0
D0701 307 1988 0 4 0
D0701 307 2117 0 3 0
D0701 307 2717 0 1 0
D0701 307 3228 0 NA 1

1.1 Observed data

N individuals = 307
N samples = 1025

1.1.1 Pomona

ggplot() +
       geom_line(data = observed.data.for.fitting.incl.neg,aes(x = time,y = titer.pomona,col = id,group = id),alpha = 0.4,size = 1,color="#9E0142") +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_classic(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 24),
             text = element_text(size = 11),
             axis.ticks = element_blank(),
             axis.text = element_blank(),
             axis.title  =  element_text(size=30,margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none")+
       xlab("Time since first positive") +
       ylab("Ab level (log2)")

ggplot() +
       geom_line(data = observed.data.for.fitting.incl.neg[1:400,],aes(x = time,y = titer.pomona,col = id,group = id),alpha = 0.4,size = 1,color="#9E0142") +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_classic(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 24),
             text = element_text(size = 11),
             axis.ticks = element_blank(),
             axis.text = element_blank(),
             axis.title  =  element_text(size=30,margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none")+
       xlab("Time since first positive") +
       ylab("Ab level (log2)")

ggplot() +
       geom_line(data = observed.data.for.fitting,aes(x = time,y = titer.pomona,col = id,group = id),alpha = 0.4,size = 0.6) +
       geom_point(data = observed.data.for.fitting,aes(x = time,y = titer.pomona,col = id,group = id),size=1.2,alpha = 0.9) +
       scale_color_gradientn(colours  =  brewer.pal(4,"Spectral")) +
       # scale_y_continuous(limits = c(2,10),breaks = 2:10) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 24),
             text = element_text(size = 11),
             axis.text = element_text(size = 10),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none")+
       xlab("Time since first positive sample") +
       ylab("Antibody level (log2) Pomona")

1.1.2 Autumnalis

ggplot() +
       geom_line(data = observed.data.for.fitting.incl.neg,aes(x = time,y = titer.aut,col = id,group = id),alpha = 0.4,size = 1,color="#9E0142") +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_classic(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 24),
             text = element_text(size = 11),
             axis.ticks = element_blank(),
             axis.text = element_blank(),
             axis.title  =  element_text(size=30,margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none")+
       xlab("Time since first positive") +
       ylab("Ab level (log2)")

ggplot() +
       geom_line(data = observed.data.for.fitting,aes(x = time,y = titer.aut,col = id,group = id),alpha = 0.4,size = 0.6) +
       geom_point(data = observed.data.for.fitting,aes(x = time,y = titer.aut,col = id,group = id),size=1.2,alpha = 0.9) +
       scale_color_gradientn(colours  =  brewer.pal(4,"Spectral")) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 24),
             text = element_text(size = 11),
             axis.text = element_text(size = 10),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none")+
       xlab("Time since first positive sample") +
       ylab("Antibody level (log2) Autumnalis")

1.1.3 Combined

plot.temp.dat = gather(observed.data.for.fitting,serovar,titer,c("titer.pomona","titer.aut"))
plot.temp.dat$id = paste0(plot.temp.dat$serovar,plot.temp.dat$id)

ggplot() +
       geom_line(data = plot.temp.dat,aes(x = time,y = titer,col = serovar,group = id),alpha = 0.3,size = 0.7) +
       #geom_point(data = observed.data.for.fitting.incl.neg,aes(x = time,y = titer.pomona,col = id,group = id),size=1.2,alpha = 0.9,col="grey") +
       #geom_point(data = observed.data.for.fitting.incl.neg,aes(x = time,y = titer.aut,col = id,group = id),size=1.2,alpha = 0.9,col="darkred") +
       #scale_color_gradientn(colours  =  brewer.pal(4,"Spectral")) +
       scale_color_manual(values = brewer.pal(11,"Spectral")[c(2,10)],labels = c("Autumnalis","Pomona"),name="Serovar") +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 24),
             text = element_text(size = 11),
             axis.text = element_text(size = 10),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  c(0.75,0.75),
             legend.background = element_rect(colour = "transparent", fill = "transparent"))+
       xlab("Time since first positive sample") +
       ylab("Antibody level (log)")

Autumnalis levels increased by 0.2 for plotting purposes.

plot.temp.dat$titer[which(plot.temp.dat$serovar=="titer.aut")] = plot.temp.dat$titer[which(plot.temp.dat$serovar=="titer.aut")] + 0.2

ggplot() +
       #geom_line(data = plot.temp.dat,aes(x = time,y = titer,col = serovar,group = id),alpha = 0.3,size = 0.7) +
       geom_point(data = plot.temp.dat,aes(x = time,y = titer,col = serovar,group = id),alpha = 0.4,size = 0.7) +
       #geom_point(data = observed.data.for.fitting.incl.neg,aes(x = time,y = titer.aut,col = id,group = id),size=1.2,alpha = 0.9,col="darkred") +
       #scale_color_gradientn(colours  =  brewer.pal(4,"Spectral")) +
       scale_color_manual(values = brewer.pal(5,"Spectral")[c(1,5)],labels = c("Autumnalis","Pomona"),name="Serovar") +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 24),
             text = element_text(size = 11),
             axis.text = element_text(size = 10),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  c(0.75,0.75),
             legend.background = element_rect(colour = "transparent", fill = "transparent"))+
       xlab("Time since first positive sample") +
       ylab("Antibody level (log)")

2 Fit model parameters

2.1 Double exponential


# set prior parameters

peak.titer.mean.prior.mean.pomona = 7
peak.titer.mean.prior.sd.pomona = 2

peak.titer.sd.prior.pomona = 2.5  # for multivariate normal prior

peak.titer.mean.prior.mean.aut = 7.5
peak.titer.mean.prior.sd.aut = 2

peak.titer.sd.prior.aut = 3  # for multivariate normal prior


decay.rate.mean.prior.shape.pomona = 1
decay.rate.mean.prior.rate.pomona = 20

decay.rate.sd.prior.shape.pomona = 2   # 7
decay.rate.sd.prior.rate.pomona = 3   # 20000

decay.rate.mean.prior.shape.aut = 1
decay.rate.mean.prior.rate.aut = 20

decay.rate.sd.prior.shape.aut = 2  # 7
decay.rate.sd.prior.rate.aut = 3   # 20000

# parameters:
# 1 = error.sd, individual (log) normal error
# 2 : 1+N.inds = time of infection, individual, as time before first positive     
# 2+N.inds : 2+N.inds+N.inds = peak titer individual

N.inds = length(unique(observed.data.for.fitting$id))

uid = unique(observed.data.for.fitting$id)

# burn.in = 40000
# iterations = 50000  # = after burn in


burn.in = 10000
iterations = 50000  # = after burn in



run.mcmc=F


if(run.mcmc==T){
       
       model1.mod = function(){
              
              # priors
              for(j in 1:length(neg.int)){
                     
                     
                     # multivariate distribution for pom and aut peak titers
                     mu_pom_aut[j,1:2] ~ dmnorm(mu_pom_aut_mean,mu_pom_aut_precision)
                     
                     # extract mean peak levels for pom and aut
                     mu_pomona[j] <- mu_pom_aut[j,1]
                     mu_aut[j] <- mu_pom_aut[j,2]
                     
                     # decay rates pom and aut
                     decay_pomona[j] ~ dnorm(decay_overall_pomona,decay_tau_overall_pomona)
                     decay_aut[j] ~ dnorm(decay_overall_aut,decay_tau_overall_aut)
                     
                     # time between peak level and first positive, shared between pom and aut
                     theta[j] ~ dunif(neg.int[j],0)
              }
              
              sigma_pomona ~ dunif(0,50)
              tau_pomona <- 1/(sigma_pomona*sigma_pomona)
              
              sigma_aut ~ dunif(0,50)
              tau_aut <- 1/(sigma_aut*sigma_aut)
              
              lab_effect ~ dnorm(0,0.01)
              
              #hyper priors
              
              # multivariate pom aut mean and sd
              mu_pom_aut_mean ~ dmnorm(mu_means,tau_means)
              mu_pom_aut_precision ~ dwish(omega,wishdf)
              
              # extract individual means peak level
              mu_overall_pomona <- mu_pom_aut_mean[1]
              mu_overall_aut <- mu_pom_aut_mean[2]
              
              # multivariate precision matrix
              inverse_mu_pom_aut_precision <- inverse(mu_pom_aut_precision)
              sigma_overall_pomona <- inverse_mu_pom_aut_precision[1,1]^(1/2)
              sigma_overall_aut <- inverse_mu_pom_aut_precision[2,2]^(1/2)
              
              # decay rates            
              
              decay_overall_pomona ~ dgamma(decay.rate.mean.prior.shape.pomona,decay.rate.mean.prior.rate.pomona)
              decay_sigma_overall_pomona ~ dgamma(decay.rate.sd.prior.shape.pomona,decay.rate.sd.prior.rate.pomona)
              decay_tau_overall_pomona <- 1/(decay_sigma_overall_pomona*decay_sigma_overall_pomona)
              
              decay_overall_aut ~ dgamma(decay.rate.mean.prior.shape.aut,decay.rate.mean.prior.rate.aut)
              decay_sigma_overall_aut ~ dgamma(decay.rate.sd.prior.shape.aut,decay.rate.sd.prior.rate.aut)
              decay_tau_overall_aut <- 1/(decay_sigma_overall_aut*decay_sigma_overall_aut)
              
              
              # likelihood
              for(i in 1:length(time)){
                     # predicted level pomona
                     titer_pred_pomona[i] <- lab_effect*lab[i] + mu_pomona[id[i]]*exp(-decay_pomona[id[i]]*(time[i]-theta[id[i]]))
                     
                     true_titer_pomona[i] ~ dnorm(titer_pred_pomona[i],tau_pomona)
                     # interval censoring
                     titer_pomona[i] ~ dinterval(true_titer_pomona[i], c(1,2,3,4,5,6,7,8,9,10,11,12,13))
                     
                     # predicted level aut
                     titer_pred_aut[i] <- lab_effect*lab[i] + mu_aut[id[i]]*exp(-decay_aut[id[i]]*(time[i]-theta[id[i]]))
                     
                     true_titer_aut[i] ~ dnorm(titer_pred_aut[i],tau_aut)
                     # interval censoring
                     titer_aut[i] ~ dinterval(true_titer_aut[i], c(1,2,3,4,5,6,7,8,9,10,11,12,13))
                     
                     # store sum loglikelihood as parameter for WAIC calculation
                     LogLik[i] = log(dnorm(titer_pomona[i],true_titer_pomona[i],tau_pomona)) + log(dnorm(titer_aut[i],true_titer_aut[i],tau_aut))
              }
       }
       
       
       model.file = "jags_model.txt"
       write.model(model1.mod,model.file)
       
       
       model.jags = jags.model(model.file,
                               data=list('titer_pomona'=observed.data.for.fitting$titer.pomona,
                                         'titer_aut'=observed.data.for.fitting$titer.aut,
                                         'time'=observed.data.for.fitting$time,
                                         'id'=observed.data.for.fitting$id,
                                         'neg.int'=neg.intervals,
                                         'lab'=observed.data.for.fitting$lab,
                                         'mu_means' = c(peak.titer.mean.prior.mean.pomona,peak.titer.mean.prior.mean.aut),
                                         'tau_means' = diag(c((1/(peak.titer.mean.prior.sd.pomona^2)),(1/(peak.titer.mean.prior.sd.aut^2)))),
                                         'omega' = diag(c((1/(peak.titer.sd.prior.pomona^2)),(1/(peak.titer.sd.prior.aut^2)))),
                                         'wishdf' = 2,
                                         'decay.rate.mean.prior.shape.pomona' = decay.rate.mean.prior.shape.pomona,
                                         'decay.rate.mean.prior.rate.pomona' = decay.rate.mean.prior.rate.pomona,
                                         "decay.rate.sd.prior.shape.pomona" = decay.rate.sd.prior.shape.pomona,
                                         "decay.rate.sd.prior.rate.pomona" = decay.rate.sd.prior.rate.pomona,
                                         'decay.rate.mean.prior.shape.aut' = decay.rate.mean.prior.shape.aut,
                                         'decay.rate.mean.prior.rate.aut' = decay.rate.mean.prior.rate.aut,
                                         "decay.rate.sd.prior.shape.aut" = decay.rate.sd.prior.shape.aut,
                                         "decay.rate.sd.prior.rate.aut" = decay.rate.sd.prior.rate.aut
                                         
                               ),
                               #inits=list('beta0'=-1),
                               n.chains=6,
                               n.adapt = 1000)
       
       
       # burn in
       update(model.jags, n.iter = burn.in, by = 1)
       
       
       
       # draw samples
       post = coda.samples(model.jags, c("mu_overall_pomona","sigma_overall_pomona","decay_overall_pomona","decay_sigma_overall_pomona","decay_pomona","mu_overall_aut","sigma_overall_aut","decay_overall_aut","decay_sigma_overall_aut","decay_aut","theta","mu_pomona","mu_aut","LogLik","mu_pom_aut_mean","mu_pom_aut_precision","lab_effect"), n.iter = iterations, thin = 1)
       
       chains.burn.df = mcmclist.to.dataframe(post)
       
       rm(post)
       
       
       # save output, needs existing MCMC_runs folder in working directory
       filename = "MCMC_antibody_decay_double_exponential.RDS"
       print(filename)
       saveRDS(chains.burn.df,filename)
       
       
       
} else {
       # load previous output if not re-running model, speeds up markdown knitting
       chains.burn.df=readRDS("MCMC_antibody_decay_double_exponential.RDS")
}


chains.burn.df.original = chains.burn.df
# only keep last 20000 iterations for each chain
chains.burn.df = chains.burn.df  %>% 
       filter(iteration > (max(iteration)-20000))

parnames = c(paste0("LogLik",1:nrow(observed.data.for.fitting)),paste0("decay.rate.aut.",uid),"decay.rate.overall.aut","decay.rate.overall.pomona",paste0("decay.rate.pomona.",uid),"decay.rate.sd.overall.aut","decay.rate.sd.overall.pomona","lab.effect",paste0("peak.titer.aut.",uid),"peak.titer.overall.aut","peak.titer.overall.pomona",paste0("mu_pom_aut_mean_pom_",1:2),paste0("mu_pom_aut_precision_",1:4),paste0("peak.titer.pomona.",uid),"peak.titer.sd.overall.aut","peak.titer.sd.overall.pomona",paste0("toi.",uid),"chain","iteration")
       
colnames(chains.burn.df) = parnames
# gelman rubin diagnostics, uncomment to run   

# gel.rub.apply.fun = function(x) gelman.fun(x = x,chains=chains.burn.df$chain,iterations = chains.burn.df$iteration)
# R.vals = gel.rub.apply.fun(chains.burn.df[,1])
# # 
# gel.rub = data.frame(variable = colnames(chains.burn.df)[-which(colnames(chains.burn.df) %in% c("chain","iteration"))],
#                      R = apply(chains.burn.df[-which(colnames(chains.burn.df) %in% c("chain","iteration"))],2,gel.rub.apply.fun))
# 
# get posterior estimates of multivariate mean and precision matrix

mnorm_means = c(mean(chains.burn.df$mu_pom_aut_mean_pom_1),mean(chains.burn.df$mu_pom_aut_mean_pom_2))

precision_matrix = matrix(data = c(
       median(chains.burn.df$mu_pom_aut_precision_1),
       median(chains.burn.df$mu_pom_aut_precision_2),
       median(chains.burn.df$mu_pom_aut_precision_3),
       median(chains.burn.df$mu_pom_aut_precision_4)),ncol=2)

2.2 Adjust time since first positive sample to estimated time since peak level

observed.data.for.fitting$time.since.peak = NA

for(i in 1:length(uid)){
       idx.current = which(observed.data.for.fitting$id == uid[i])
       peak.titer.time = round(max.dens.fun(x = chains.burn.df[,paste0("toi.",i)],neg.interval = neg.intervals[i]))
       observed.data.for.fitting$time.since.peak[idx.current] = observed.data.for.fitting$time[idx.current] - peak.titer.time
}

2.3 Single exponential

if(run.mcmc==T){
       
       
       # set prior parameters
       
       peak.titer.mean.prior.mean.pomona = 7.5
       peak.titer.mean.prior.sd.pomona = 0.7
       
       peak.titer.sd.prior.pomona = 3  # for multivariate normal prior
       
       peak.titer.mean.prior.mean.aut = 7
       peak.titer.mean.prior.sd.aut = 0.7
       
       peak.titer.sd.prior.aut = 3  # for multivariate normal prior
       
       
       decay.rate.mean.prior.mean.pomona = 0.0009
       decay.rate.mean.prior.sd.pomona = 0.0001
       decay.rate.sd.prior.shape.pomona = 7
       decay.rate.sd.prior.rate.pomona = 20000
       
       decay.rate.mean.prior.mean.aut = 0.0009
       decay.rate.mean.prior.sd.aut = 0.0001
       decay.rate.sd.prior.shape.aut = 7
       decay.rate.sd.prior.rate.aut = 20000
       
       # parameters:
       # 1 = error.sd, individual (log) normal error
       # 2 : 1+N.inds = time of infection, individual, as time before first positive     
       # 2+N.inds : 2+N.inds+N.inds = peak titer individual
       
       N.inds = length(unique(observed.data.for.fitting$id))
       
       uid = unique(observed.data.for.fitting$id)
       
       burn.in = 40000
       iterations = 50000  # = after burn in
       
       
       model1.mod = function(){
              
              # priors
              for(j in 1:length(neg.int)){
                     
                     
                     # multivariate distribution for pom and aut peak titers
                     mu_pom_aut[j,1:2] ~ dmnorm(mu_pom_aut_mean,mu_pom_aut_precision)
                     
                     # extract mean peak levels for pom and aut
                     mu_pomona[j] <- mu_pom_aut[j,1]
                     mu_aut[j] <- mu_pom_aut[j,2]
                     
                     # decay rates pom and aut
                     decay_pomona[j] ~ dnorm(decay_overall_pomona,decay_tau_overall_pomona)
                     decay_aut[j] ~ dnorm(decay_overall_aut,decay_tau_overall_aut)
                     
                     # time between peak level and first positive, shared between pom and aut
                     theta[j] ~ dunif(neg.int[j],0)
              }
              
              sigma_pomona ~ dunif(0,50)
              tau_pomona <- 1/(sigma_pomona*sigma_pomona)
              
              sigma_aut ~ dunif(0,50)
              tau_aut <- 1/(sigma_aut*sigma_aut)
              
              lab_effect ~ dnorm(0,0.01)
              
              #hyper priors
              
              # multivariate pom aut mean and sd
              mu_pom_aut_mean ~ dmnorm(mu_means,tau_means)
              mu_pom_aut_precision ~ dwish(omega,wishdf)
              
              # extract individual means peak level
              mu_overall_pomona <- mu_pom_aut_mean[1]
              mu_overall_aut <- mu_pom_aut_mean[2]
              
              # multivariate precision matrix
              inverse_mu_pom_aut_precision <- inverse(mu_pom_aut_precision)
              sigma_overall_pomona <- inverse_mu_pom_aut_precision[1,1]^(1/2)
              sigma_overall_aut <- inverse_mu_pom_aut_precision[2,2]^(1/2)
              
              # decay rates            
              decay_overall_pomona ~ dnorm(decay.rate.mean.prior.mean.pomona,1/(decay.rate.mean.prior.sd.pomona*decay.rate.mean.prior.sd.pomona))
              decay_sigma_overall_pomona ~ dgamma(decay.rate.sd.prior.shape.pomona,decay.rate.sd.prior.rate.pomona)
              decay_tau_overall_pomona <- 1/(decay_sigma_overall_pomona*decay_sigma_overall_pomona)
              
              decay_overall_aut ~ dnorm(decay.rate.mean.prior.mean.aut,1/(decay.rate.mean.prior.sd.aut*decay.rate.mean.prior.sd.aut))
              decay_sigma_overall_aut ~ dgamma(decay.rate.sd.prior.shape.aut,decay.rate.sd.prior.rate.aut)
              decay_tau_overall_aut <- 1/(decay_sigma_overall_aut*decay_sigma_overall_aut)
              
              
              # likelihood
              for(i in 1:length(time)){
                     # predicted level pomona
                     
                     titer_pred_pomona[i] <- lab_effect*lab[i] +mu_pomona[id[i]] - decay_pomona[id[i]]*(time[i]-theta[id[i]])
                     true_titer_pomona[i] ~ dnorm(titer_pred_pomona[i],tau_pomona)
                     # interval censoring
                     titer_pomona[i] ~ dinterval(true_titer_pomona[i], c(1,2,3,4,5,6,7,8,9,10,11,12,13))
                     
                     # predicted level aut
                     titer_pred_aut[i] <- lab_effect*lab[i] +mu_aut[id[i]] - decay_aut[id[i]]*(time[i]-theta[id[i]])
                     true_titer_aut[i] ~ dnorm(titer_pred_aut[i],tau_aut)
                     # interval censoring
                     titer_aut[i] ~ dinterval(true_titer_aut[i], c(1,2,3,4,5,6,7,8,9,10,11,12,13))
                     
                     # store sum loglikelihood as parameter for WAIC calculation
                     LogLik[i] = log(dnorm(titer_pomona[i],true_titer_pomona[i],tau_pomona)) + log(dnorm(titer_aut[i],true_titer_aut[i],tau_aut))
              }
       }
       
       
       model.file = "jags_model.txt"
       write.model(model1.mod,model.file)
       
       
       model.jags = jags.model(model.file,
                               data=list('titer_pomona'=observed.data.for.fitting$titer.pomona,
                                         'titer_aut'=observed.data.for.fitting$titer.aut,
                                         'time'=observed.data.for.fitting$time,
                                         'id'=observed.data.for.fitting$id,
                                         'lab'=observed.data.for.fitting$lab,
                                         'neg.int'=neg.intervals,
                                         'mu_means' = c(peak.titer.mean.prior.mean.pomona,peak.titer.mean.prior.mean.aut),
                                         'tau_means' = diag(c((1/(peak.titer.mean.prior.sd.pomona^2)),(1/(peak.titer.mean.prior.sd.aut^2)))),
                                         'omega' = diag(c((1/(peak.titer.sd.prior.pomona^2)),(1/(peak.titer.sd.prior.aut^2)))),
                                         'wishdf' = 2,
                                         "decay.rate.mean.prior.mean.pomona" = decay.rate.mean.prior.mean.pomona,
                                         "decay.rate.mean.prior.sd.pomona" = decay.rate.mean.prior.sd.pomona,
                                         "decay.rate.sd.prior.shape.pomona" = decay.rate.sd.prior.shape.pomona,
                                         "decay.rate.sd.prior.rate.pomona" = decay.rate.sd.prior.rate.pomona,
                                         "decay.rate.mean.prior.mean.aut" = decay.rate.mean.prior.mean.aut,
                                         "decay.rate.mean.prior.sd.aut" = decay.rate.mean.prior.sd.aut,
                                         "decay.rate.sd.prior.shape.aut" = decay.rate.sd.prior.shape.aut,
                                         "decay.rate.sd.prior.rate.aut" = decay.rate.sd.prior.rate.aut
                                         
                               ),
                               #inits=list('beta0'=-1),
                               n.chains=6,
                               n.adapt = 2500)
       
       
       # burn in
       update(model.jags, n.iter = burn.in, by = 1)
       
       
       
       # draw samples
       post = coda.samples(model.jags, c("mu_overall_pomona","sigma_overall_pomona","decay_overall_pomona","decay_sigma_overall_pomona","decay_pomona","mu_overall_aut","sigma_overall_aut","decay_overall_aut","decay_sigma_overall_aut","decay_aut","theta","mu_pomona","mu_aut","LogLik","mu_pom_aut_mean","lab_effect"), n.iter = iterations, thin = 10)
       
       chains.burn.df = mcmclist.to.dataframe(post)
       
       # parnames = c(paste0("LogLik",1:nrow(observed.data.for.fitting)),paste0("decay.rate.aut.",uid),"decay.rate.overall.aut","decay.rate.overall.pomona",paste0("decay.rate.pomona.",uid),"decay.rate.sd.overall.aut","decay.rate.sd.overall.pomona",paste0("peak.titer.aut.",uid),"peak.titer.overall.aut","peak.titer.overall.pomona",paste0("mu_pom_aut_mean_pom_",1:2),paste0("mu_pom_aut_precision_",1:4),paste0("peak.titer.pomona.",uid),"peak.titer.sd.overall.aut","peak.titer.sd.overall.pomona",paste0("toi.",uid),"chain","iteration")
       # 
       # colnames(chains.burn.df) = parnames
       
       
       # save output, needs existing MCMC_runs folder in working directory
       filename = "MCMC_antibody_decay_single_exponential.RDS"
       print(filename)
       saveRDS(chains.burn.df,filename)
       
       
       
} else {
       # load previous output if not re-running model, speeds up markdown knitting
       chains.burn.df.1=readRDS("MCMC_antibody_decay_single_exponential.RDS")
}

2.4 Power

if(run.mcmc==T){
       
       
       # set prior parameters
       
       peak.titer.mean.prior.mean.pomona = 7.5
       peak.titer.mean.prior.sd.pomona = 0.7
       
       peak.titer.sd.prior.pomona = 3  # for multivariate normal prior
       
       peak.titer.mean.prior.mean.aut = 7
       peak.titer.mean.prior.sd.aut = 0.7
       
       peak.titer.sd.prior.aut = 3  # for multivariate normal prior
       
       
       shape.sd.prior = 3
       scale.sd.prior = 0.003
       
       shape.mean.sd.prior = 1
       scale.mean.sd.prior = 0.0005
       
       
       log.mean.shape.prior.mean = log(0.5)
       log.mean.shape.prior.sd = 0.1
       log.mean.scale.prior.mean = log(0.0005)
       log.mean.scale.prior.sd = 1
       
       log.sd.shape.prior.shape = 2
       log.sd.shape.prior.rate = 0.75
       log.sd.scale.prior.shape = 2
       log.sd.scale.prior.rate = 0.5
       
       
       
       
       # parameters:
       # 1 = error.sd, individual (log) normal error
       # 2 : 1+N.inds = time of infection, individual, as time before first positive     
       # 2+N.inds : 2+N.inds+N.inds = peak titer individual
       
       N.inds = length(unique(observed.data.for.fitting$id))
       
       uid = unique(observed.data.for.fitting$id)
       
       burn.in = 40000
       iterations = 50000  # = after burn in
       
       
       
       model1.mod = function(){
              
              # priors
              for(j in 1:length(neg.int)){
                     
                     
                     # multivariate distribution for pom and aut peak titers
                     mu_pom_aut[j,1:2] ~ dmnorm(mu_pom_aut_mean,mu_pom_aut_precision)
                     
                     # extract mean peak levels for pom and aut
                     mu_pomona[j] <- mu_pom_aut[j,1]
                     exp.mu_pomona[j] <- 100*2^(mu_pomona[j]-1)
                     mu_aut[j] <- mu_pom_aut[j,2]
                     exp.mu_aut[j] <- 100*2^(mu_aut[j]-1)
                     
                     # decay parameters pom and aut
                     log_shape_scale_pomona[j,1:2] ~ dmnorm(shape_scale_pomona_mean,shape_scale_pomona_precision)
                     log.shape_pomona[j] <- log_shape_scale_pomona[j,1]
                     log.scale_pomona[j] <- log_shape_scale_pomona[j,2]
                     
                     shape_pomona[j] <- exp(log.shape_pomona[j]) + 1
                     scale_pomona[j] <- exp(log.scale_pomona[j])
                     
                     log_shape_scale_aut[j,1:2] ~ dmnorm(shape_scale_aut_mean,shape_scale_aut_precision)
                     log.shape_aut[j] <- log_shape_scale_aut[j,1]
                     log.scale_aut[j] <- log_shape_scale_aut[j,2]
                     
                     shape_aut[j] <- exp(log.shape_aut[j]) + 1
                     scale_aut[j] <- exp(log.scale_aut[j])
                     
                     
                     # time between peak level and first positive, shared between pom and aut
                     theta[j] ~ dunif(neg.int[j],0)
              }
              
              
              
              lab_effect ~ dnorm(0,0.01)
              
              sigma_pomona ~ dunif(0,50)
              tau_pomona <- 1/(sigma_pomona*sigma_pomona)
              
              sigma_aut ~ dunif(0,50)
              tau_aut <- 1/(sigma_aut*sigma_aut)
              
              
              #hyper priors
              
              # multivariate pom aut mean and sd
              mu_pom_aut_mean ~ dmnorm(mu_means,tau_means)
              mu_pom_aut_precision ~ dwish(omega,wishdf)
              
              # extract individual means peak level
              mu_overall_pomona <- mu_pom_aut_mean[1]
              mu_overall_aut <- mu_pom_aut_mean[2]
              
              # multivariate precision matrix
              inverse_mu_pom_aut_precision <- inverse(mu_pom_aut_precision)
              sigma_overall_pomona <- inverse_mu_pom_aut_precision[1,1]^(1/2)
              sigma_overall_aut <- inverse_mu_pom_aut_precision[2,2]^(1/2)
              
              
              
              # multivariate decay rate parameters pomona
              shape_scale_pomona_mean ~ dmnorm(mean_shape_scale_pomona_mean,mean_shape_scale_pomona_tau)
              shape_scale_pomona_precision ~ dwish(omega_shape_scale_pomona,wishdf_shape_scale_pomona)
              
              # extract individual means decay rate parameters pomona
              mu_overall_shape_pomona <- exp(shape_scale_pomona_mean[1])+1
              mu_overall_scale_pomona <- exp(shape_scale_pomona_mean[2])
              
              # multivariate precision matrix pomona
              inverse_mu_shape_scale_pomona_precision <- inverse(shape_scale_pomona_precision)
              sigma_overall_shape_pomona <- inverse_mu_shape_scale_pomona_precision[1,1]^(1/2)
              sigma_overall_scale_pomona <- inverse_mu_shape_scale_pomona_precision[2,2]^(1/2)
              
              
              
              # multivariate decay rate parameters aut
              shape_scale_aut_mean ~ dmnorm(mean_shape_scale_aut_mean,mean_shape_scale_aut_tau)
              shape_scale_aut_precision ~ dwish(omega_shape_scale_aut,wishdf_shape_scale_aut)
              
              # extract individual means decay rate parameters aut
              mu_overall_shape_aut <- exp(shape_scale_aut_mean[1])+1
              mu_overall_scale_aut <- exp(shape_scale_aut_mean[2])
              
              # multivariate precision matrix aut
              inverse_mu_shape_scale_aut_precision <- inverse(shape_scale_aut_precision)
              sigma_overall_shape_aut <- inverse_mu_shape_scale_aut_precision[1,1]^(1/2)
              sigma_overall_scale_aut <- inverse_mu_shape_scale_aut_precision[2,2]^(1/2)
              
              
              
              
              # likelihood
              for(i in 1:length(time)){
                     # predicted level pomona
                     titer_pred_pomona[i] <-  exp.mu_pomona[id[i]]*(1+(shape_pomona[id[i]]-1)*(exp.mu_pomona[id[i]]^(shape_pomona[id[i]]-1))*scale_pomona[id[i]]*(time[i]-theta[id[i]]))^(-1/(shape_pomona[id[i]]-1))
                     log2_titer_pred_pomona[i] <- 1+(log(titer_pred_pomona[i]/100)/log(2))
                     lab_effect_log2_titer_pred_pomona[i] <- lab_effect*lab[i] + log2_titer_pred_pomona[i]
                     true_titer_pomona[i] ~ dnorm(lab_effect_log2_titer_pred_pomona[i],tau_pomona)
                     # interval censoring
                     titer_pomona[i] ~ dinterval(true_titer_pomona[i], c(1,2,3,4,5,6,7,8,9,10,11,12,13))
                     
                     
                     titer_pred_aut[i] <-  exp.mu_aut[id[i]]*(1+(shape_aut[id[i]]-1)*(exp.mu_aut[id[i]]^(shape_aut[id[i]]-1))*scale_aut[id[i]]*(time[i]-theta[id[i]]))^(-1/(shape_aut[id[i]]-1))
                     log2_titer_pred_aut[i] <- 1+(log(titer_pred_aut[i]/100)/log(2))
                     lab_effect_log2_titer_pred_aut[i] <- lab_effect*lab[i] + log2_titer_pred_aut[i]
                     true_titer_aut[i] ~ dnorm(lab_effect_log2_titer_pred_aut[i],tau_aut)
                     # interval censoring
                     titer_aut[i] ~ dinterval(true_titer_aut[i], c(1,2,3,4,5,6,7,8,9,10,11,12,13))
                     
                     
                     # store sum loglikelihood as parameter for WAIC calculation
                     LogLik[i] = log(dnorm(titer_pomona[i],true_titer_pomona[i],tau_pomona)) + log(dnorm(titer_aut[i],true_titer_aut[i],tau_aut))
              }
       }
       
       
       
       model2.uncorrelated_shape_scale.mod = function(){
              
              # priors
              for(j in 1:length(neg.int)){
                     
                     
                     # multivariate distribution for pom and aut peak titers
                     mu_pom_aut[j,1:2] ~ dmnorm(mu_pom_aut_mean,mu_pom_aut_precision)
                     
                     # extract mean peak levels for pom and aut
                     mu_pomona[j] <- mu_pom_aut[j,1]
                     exp.mu_pomona[j] <- 100*2^(mu_pomona[j]-1)
                     mu_aut[j] <- mu_pom_aut[j,2]
                     exp.mu_aut[j] <- 100*2^(mu_aut[j]-1)
                     
                     # decay parameters pom and aut
                     log.shape_pomona[j] ~ dnorm(log_shape_pomona_overall_mean,log_shape_pomona_overall_tau)
                     log.scale_pomona[j] ~ dnorm(log_scale_pomona_overall_mean,log_scale_pomona_overall_tau)
                     
                     shape_pomona[j] <- exp(log.shape_pomona[j]) + 1
                     scale_pomona[j] <- exp(log.scale_pomona[j])
                     
                     log.shape_aut[j] ~ dnorm(log_shape_aut_overall_mean,log_shape_aut_overall_tau)
                     log.scale_aut[j] ~ dnorm(log_scale_aut_overall_mean,log_scale_aut_overall_tau)
                     
                     shape_aut[j] <- exp(log.shape_aut[j]) + 1
                     scale_aut[j] <- exp(log.scale_aut[j])
                     
                     
                     # time between peak level and first positive, shared between pom and aut
                     theta[j] ~ dunif(neg.int[j],0)
              }
              
              
              
              lab_effect ~ dnorm(0,0.01)
              
              sigma_pomona ~ dunif(0,50)
              tau_pomona <- 1/(sigma_pomona*sigma_pomona)
              
              sigma_aut ~ dunif(0,50)
              tau_aut <- 1/(sigma_aut*sigma_aut)
              
              
              #hyper priors
              
              # multivariate pom aut mean and sd
              mu_pom_aut_mean ~ dmnorm(mu_means,tau_means)
              mu_pom_aut_precision ~ dwish(omega,wishdf)
              
              # extract serovar peak level means
              mu_overall_pomona <- mu_pom_aut_mean[1]
              mu_overall_aut <- mu_pom_aut_mean[2]
              
              # multivariate precision matrix
              inverse_mu_pom_aut_precision <- inverse(mu_pom_aut_precision)
              sigma_overall_pomona <- inverse_mu_pom_aut_precision[1,1]^(1/2)
              sigma_overall_aut <- inverse_mu_pom_aut_precision[2,2]^(1/2)
              
              
              
              # decay rate parameters pomona
              log_shape_pomona_overall_mean ~ dnorm(log_shape_pomona_overall_mean_prior_mean,1/(log_shape_pomona_overall_mean_prior_sd*log_shape_pomona_overall_mean_prior_sd))
              
              log_shape_pomona_overall_sd ~ dgamma(log_shape_pomona_overall_sd_prior_shape,log_shape_pomona_overall_sd_prior_rate)
              log_shape_pomona_overall_tau <- 1/(log_shape_pomona_overall_sd*log_shape_pomona_overall_sd)
              
              shape_pomona_overall_mean <- exp(log_shape_pomona_overall_mean) + 1
              
              
              log_scale_pomona_overall_mean ~ dnorm(log_scale_pomona_overall_mean_prior_mean,1/(log_scale_pomona_overall_mean_prior_sd*log_scale_pomona_overall_mean_prior_sd))
              
              log_scale_pomona_overall_sd ~ dgamma(log_scale_pomona_overall_sd_prior_shape,log_scale_pomona_overall_sd_prior_rate)
              log_scale_pomona_overall_tau <- 1/(log_scale_pomona_overall_sd*log_scale_pomona_overall_sd)
              
              scale_pomona_overall_mean <- exp(log_scale_pomona_overall_mean)
              
              
              
              
              # decay rate parameters aut
              log_shape_aut_overall_mean ~ dnorm(log_shape_aut_overall_mean_prior_mean,1/(log_shape_aut_overall_mean_prior_sd*log_shape_aut_overall_mean_prior_sd))
              
              log_shape_aut_overall_sd ~ dgamma(log_shape_aut_overall_sd_prior_shape,log_shape_aut_overall_sd_prior_rate)
              log_shape_aut_overall_tau <- 1/(log_shape_aut_overall_sd*log_shape_aut_overall_sd)
              
              shape_aut_overall_mean <- exp(log_shape_aut_overall_mean) + 1
              
              
              log_scale_aut_overall_mean ~ dnorm(log_scale_aut_overall_mean_prior_mean,1/(log_scale_aut_overall_mean_prior_sd*log_scale_aut_overall_mean_prior_sd))
              
              log_scale_aut_overall_sd ~ dgamma(log_scale_aut_overall_sd_prior_shape,log_scale_aut_overall_sd_prior_rate)
              log_scale_aut_overall_tau <- 1/(log_scale_aut_overall_sd*log_scale_aut_overall_sd)
              
              scale_aut_overall_mean <- exp(log_scale_aut_overall_mean)
              
              
              
              
              
              
              # likelihood
              for(i in 1:length(time)){
                     
                     # predicted level pomona
                     titer_pred_pomona[i] <-  exp.mu_pomona[id[i]]*(1+(shape_pomona[id[i]]-1)*(exp.mu_pomona[id[i]]^(shape_pomona[id[i]]-1))*scale_pomona[id[i]]*(time[i]-theta[id[i]]))^(-1/(shape_pomona[id[i]]-1))
                     log2_titer_pred_pomona[i] <- 1+(log(titer_pred_pomona[i]/100)/log(2))
                     lab_effect_log2_titer_pred_pomona[i] <- lab_effect*lab[i] + log2_titer_pred_pomona[i]
                     true_titer_pomona[i] ~ dnorm(lab_effect_log2_titer_pred_pomona[i],tau_pomona)
                     # interval censoring
                     titer_pomona[i] ~ dinterval(true_titer_pomona[i], c(1,2,3,4,5,6,7,8,9,10,11,12,13))
                     
                     
                     titer_pred_aut[i] <-  exp.mu_aut[id[i]]*(1+(shape_aut[id[i]]-1)*(exp.mu_aut[id[i]]^(shape_aut[id[i]]-1))*scale_aut[id[i]]*(time[i]-theta[id[i]]))^(-1/(shape_aut[id[i]]-1))
                     log2_titer_pred_aut[i] <- 1+(log(titer_pred_aut[i]/100)/log(2))
                     lab_effect_log2_titer_pred_aut[i] <- lab_effect*lab[i] + log2_titer_pred_aut[i]
                     true_titer_aut[i] ~ dnorm(lab_effect_log2_titer_pred_aut[i],tau_aut)
                     # interval censoring
                     titer_aut[i] ~ dinterval(true_titer_aut[i], c(1,2,3,4,5,6,7,8,9,10,11,12,13))
                     
                     
                     # store sum loglikelihood as parameter for WAIC calculation
                     LogLik[i] = log(dnorm(titer_pomona[i],true_titer_pomona[i],tau_pomona)) + log(dnorm(titer_aut[i],true_titer_aut[i],tau_aut))
              }
       }
       
       
       
       
       model.file = "jags_model.txt"
       write.model(model2.uncorrelated_shape_scale.mod,model.file)
       
       
       model.jags = jags.model(model.file,
                               data=list('titer_pomona'=observed.data.for.fitting$titer.pomona,
                                         'titer_aut'=observed.data.for.fitting$titer.aut,
                                         'time'=observed.data.for.fitting$time,
                                         'id'=observed.data.for.fitting$id,
                                         'neg.int'=neg.intervals,
                                         'lab'=observed.data.for.fitting$lab,
                                         'mu_means' = c(peak.titer.mean.prior.mean.pomona,peak.titer.mean.prior.mean.aut),
                                         'tau_means' = diag(c((1/(peak.titer.mean.prior.sd.pomona^2)),(1/(peak.titer.mean.prior.sd.aut^2)))),
                                         'omega' = diag(c((1/(peak.titer.sd.prior.pomona^2)),(1/(peak.titer.sd.prior.aut^2)))),
                                         'wishdf' = 2,
                                         'log_shape_pomona_overall_mean_prior_mean' = log.mean.shape.prior.mean,
                                         'log_shape_pomona_overall_mean_prior_sd' = log.mean.shape.prior.sd,
                                         'log_scale_pomona_overall_mean_prior_mean' = log.mean.scale.prior.mean,
                                         'log_scale_pomona_overall_mean_prior_sd' = log.mean.scale.prior.sd,
                                         'log_shape_aut_overall_mean_prior_mean' = log.mean.shape.prior.mean,
                                         'log_shape_aut_overall_mean_prior_sd' = log.mean.shape.prior.sd,
                                         'log_scale_aut_overall_mean_prior_mean' = log.mean.scale.prior.mean,
                                         'log_scale_aut_overall_mean_prior_sd' = log.mean.scale.prior.sd,
                                         'log_shape_pomona_overall_sd_prior_shape' = log.sd.shape.prior.shape,
                                         'log_shape_pomona_overall_sd_prior_rate' = log.sd.shape.prior.rate,
                                         'log_scale_pomona_overall_sd_prior_shape' = log.sd.scale.prior.shape,
                                         'log_scale_pomona_overall_sd_prior_rate' = log.sd.scale.prior.rate,
                                         'log_shape_aut_overall_sd_prior_shape' = log.sd.shape.prior.shape,
                                         'log_shape_aut_overall_sd_prior_rate' = log.sd.shape.prior.rate,
                                         'log_scale_aut_overall_sd_prior_shape' = log.sd.scale.prior.shape,
                                         'log_scale_aut_overall_sd_prior_rate' = log.sd.scale.prior.rate
                               ),
                               #inits=list('beta0'=-1),
                               n.chains=6,
                               n.adapt = 5000)
       
       
       # burn in
       update(model.jags, n.iter = burn.in, by = 1)
       
       
       
       # draw samples
       
       post = coda.samples(model.jags, c("mu_overall_pomona","sigma_overall_pomona","mu_overall_aut","sigma_overall_aut","theta","LogLik","mu_pomona","mu_aut","mu_pom_aut_mean","shape_pomona_overall_mean","scale_pomona_overall_mean","shape_aut_overall_mean","scale_aut_overall_mean","shape_pomona","shape_aut","scale_pomona","scale_aut","log_shape_pomona_overall_sd","log_scale_pomona_overall_sd","log_shape_aut_overall_sd","log_scale_aut_overall_sd","lab_effect"), n.iter = iterations, thin = 10)
       
       
       
       chains.burn.df = mcmclist.to.dataframe(post)
       
       
       # save output, needs existing MCMC_runs folder in working directory
       filename = "MCMC_antibody_decay_power"
       print(filename)
       saveRDS(chains.burn.df,filename)
       
       
       
} else {
       # load previous output if not re-running model, speeds up markdown knitting
       chains.burn.df.3=readRDS("MCMC_antibody_decay_power.RDS")
}

2.5 Gather output three functions

if(run.mcmc==T) {
       

       chains.burn.df.2 = chains.burn.df.original
       
       N.inds = length(grep("theta",names(chains.burn.df.1)))
       
       # get posterior estimates     
       
       
       post.overall = data.frame(model = 1:3,
                                 RMSE = NA,
                                 WAIC = NA,
                                 LOOIC = NA,
                                 peak.titer.mean = NA,
                                 peak.titer.aut.mean = NA,
                                 peak.titer.mean.95lo = NA,
                                 peak.titer.mean.95hi = NA,
                                 peak.titer.sd = NA,
                                 peak.titer.sd.95lo = NA,
                                 peak.titer.sd.95hi = NA,
                                 decay.mean = NA,
                                 decay.aut.mean = NA,
                                 decay.mean.95lo = NA,
                                 decay.mean.95hi = NA,
                                 shape.mean = NA,
                                 shape.mean.95lo = NA,
                                 shape.mean.95hi = NA,
                                 scale.mean = NA,
                                 scale.mean.95lo = NA,
                                 scale.mean.95hi = NA
       )
       
       
       
       
       
       
       
       
       post.individual = data.frame(model = rep(1:3,each = N.inds),
                                    id = rep(1:N.inds,3),
                                    toi.mean = NA,
                                    toi.mean.95lo = NA,
                                    toi.mean.95hi = NA,
                                    toi.95.information.gained = NA,
                                    toi.mean.50lo = NA,
                                    toi.mean.50hi = NA,
                                    toi.50.information.gained = NA,
                                    peak.titer.mean = NA,
                                    peak.titer.mean.95lo = NA,
                                    peak.titer.mean.95hi = NA,
                                    decay.rate.mean = NA,
                                    decay.rate.mean.95lo = NA,
                                    decay.rate.mean.95hi = NA,
                                    shape.mean = NA,
                                    shape.mean.95lo = NA,
                                    shape.mean.95hi = NA,
                                    scale.mean = NA,
                                    scale.mean.95lo = NA,
                                    scale.mean.95hi = NA
       )
       
       
       
       models = 1:3
       
       for(model in 1:length(models)){
              
              chains.burn.df.current = get(paste0("chains.burn.df.",model))
              
              
              # pop level parameters    
              
              chains.burn.df.loglik = chains.burn.df.current[,grep("LogLik",colnames(chains.burn.df.current))]
              
              chains.burn.df.loglik = chains.burn.df.loglik[is.finite(rowSums(chains.burn.df.loglik)),]
              
              loglik.all.matrix = as.matrix(chains.burn.df.loglik)
              rm(chains.burn.df.loglik)
              
              WAIC.val = waic(loglik.all.matrix)
              post.overall[model,"WAIC"] = round(WAIC.val$estimates["waic",][1],1)
              LOOIC.val = loo(loglik.all.matrix)
              post.overall[model,"LOOIC"] = round(LOOIC.val$estimates["looic",][1],1)
              
              
              dens = density(chains.burn.df.current$mu_overall_pomona)
              hpd.int.peak.titer = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
              peak.titer.overall.mean = mean(chains.burn.df.current$mu_overall_pomona)
              
              post.overall[model,"peak.titer.mean"] = peak.titer.overall.mean
              post.overall[model,"peak.titer.mean.95lo"] = hpd.int.peak.titer[1]
              post.overall[model,"peak.titer.mean.95hi"] = hpd.int.peak.titer[2]
              
              dens = density(chains.burn.df.current$mu_overall_aut)
              peak.titer.overall.mean = mean(chains.burn.df.current$mu_overall_aut)
              
              post.overall[model,"peak.titer.aut.mean"] = peak.titer.overall.mean
              
              
              dens = density(chains.burn.df.current$sigma_overall_pomona)
              hpd.int.peak.titer.sd = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
              peak.titer.sd.overall.mean = mean(chains.burn.df.current$sigma_overall_pomona)
              
              post.overall[model,"peak.titer.sd"] = peak.titer.sd.overall.mean
              post.overall[model,"peak.titer.sd.95lo"] = hpd.int.peak.titer.sd[1]
              post.overall[model,"peak.titer.sd.95hi"] = hpd.int.peak.titer.sd[2]
              
              if(model %in% c(1,2)) {
                     dens = density(chains.burn.df.current$decay_overall_pomona)
                     hpd.int.decay = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
                     decay.overall.mean = mean(chains.burn.df.current$decay_overall_pomona)
                     
                     post.overall[model,"decay.mean"] = decay.overall.mean
                     post.overall[model,"decay.mean.95lo"] = hpd.int.decay[1]
                     post.overall[model,"decay.mean.95hi"] = hpd.int.decay[2]
                     
                     dens = density(chains.burn.df.current$decay_overall_aut)
                     hpd.int.decay = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
                     decay.overall.mean = mean(chains.burn.df.current$decay_overall_aut)
                     
                     post.overall[model,"decay.aut.mean"] = decay.overall.mean
                     
              } else {
                     
                     dens = density(chains.burn.df.current$shape_pomona_overall_mean)
                     hpd.int.shape = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
                     shape.overall.mean = mean(chains.burn.df.current$shape_pomona_overall_mean)
                     
                     post.overall[model,"shape.mean"] = shape.overall.mean
                     post.overall[model,"shape.mean.95lo"] = hpd.int.shape[1]
                     post.overall[model,"shape.mean.95hi"] = hpd.int.shape[2]   
                     
                     
                     dens = density(chains.burn.df.current$scale_pomona_overall_mean)
                     hpd.int.scale = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
                     scale.overall.mean = mean(chains.burn.df.current$scale_pomona_overall_mean)
                     
                     post.overall[model,"scale.mean"] = scale.overall.mean
                     post.overall[model,"scale.mean.95lo"] = hpd.int.scale[1]
                     post.overall[model,"scale.mean.95hi"] = hpd.int.scale[2]   
                     
              }
              
              
              
              # individual level parameters   
              
              for(i in 1:N.inds){
                     
                     
                     post.individual[which(post.individual$id==i & post.individual$model==model),"toi.mean"] = max.dens.fun(chains.burn.df.current[,paste0("theta[",i,"]")])
                     
                     dens = density(chains.burn.df.current[,paste0("theta[",i,"]")])
                     hpd.int.toi = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
                     
                     post.individual[which(post.individual$id==i & post.individual$model==model),"toi.mean.95lo"] = hpd.int.toi[1]
                     post.individual[which(post.individual$id==i & post.individual$model==model),"toi.mean.95hi"] = hpd.int.toi[2]
                     post.individual[which(post.individual$id==i & post.individual$model==model),"toi.95.information.gained"] = round(1-(hpd.int.toi[2]-hpd.int.toi[1])/abs(neg.intervals[i]),3)
                     
                     
                     
                     hpd.int.toi = HDInterval::hdi(dens,credMass = 0.5,allowSplit=F)  
                     post.individual[which(post.individual$id==i & post.individual$model==model),"toi.mean.50lo"] = hpd.int.toi[1]
                     post.individual[which(post.individual$id==i & post.individual$model==model),"toi.mean.50hi"] = hpd.int.toi[2]
                     post.individual[which(post.individual$id==i & post.individual$model==model),"toi.50.information.gained"] = round(1-(hpd.int.toi[2]-hpd.int.toi[1])/abs(neg.intervals[i]),3)
                     
                     
                     post.individual[which(post.individual$id==i & post.individual$model==model),"peak.titer.mean"] = mean(chains.burn.df.current[,paste0("mu_pomona[",i,"]")])
                     
                     dens = density(chains.burn.df.current[,paste0("mu_pomona[",i,"]")])
                     hpd.int.peak.titer = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
                     
                     post.individual[which(post.individual$id==i & post.individual$model==model),"peak.titer.mean.95lo"] = hpd.int.peak.titer[1]
                     post.individual[which(post.individual$id==i & post.individual$model==model),"peak.titer.mean.95hi"] = hpd.int.peak.titer[2]
                     
                     
                     if(model %in% 1:2) {
                            post.individual[which(post.individual$id==i & post.individual$model==model),"decay.rate.mean"] = mean(chains.burn.df.current[,paste0("decay_pomona[",i,"]")])
                            
                            dens = density(chains.burn.df.current[,paste0("decay_pomona[",i,"]")])
                            hpd.int.peak.titer = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
                            
                            post.individual[which(post.individual$id==i & post.individual$model==model),"decay.rate.mean.95lo"] = hpd.int.peak.titer[1]
                            post.individual[which(post.individual$id==i & post.individual$model==model),"decay.rate.mean.95hi"] = hpd.int.peak.titer[2]
                     }
                     
                     
                     if(model == 3) {
                            post.individual[which(post.individual$id==i & post.individual$model==model),"shape.mean"] = mean(chains.burn.df.current[,paste0("shape_pomona[",i,"]")])
                            
                            dens = density(chains.burn.df.current[,paste0("shape_pomona[",i,"]")])
                            hpd.int.peak.titer = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
                            
                            post.individual[which(post.individual$id==i & post.individual$model==model),"shape.mean.95lo"] = hpd.int.peak.titer[1]
                            post.individual[which(post.individual$id==i & post.individual$model==model),"shape.mean.95hi"] = hpd.int.peak.titer[2]
                            
                            post.individual[which(post.individual$id==i & post.individual$model==model),"scale.mean"] = mean(chains.burn.df.current[,paste0("scale_pomona[",i,"]")])
                            
                            dens = density(chains.burn.df.current[,paste0("scale_pomona[",i,"]")])
                            hpd.int.peak.titer = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
                            
                            post.individual[which(post.individual$id==i & post.individual$model==model),"scale.mean.95lo"] = hpd.int.peak.titer[1]
                            post.individual[which(post.individual$id==i & post.individual$model==model),"scale.mean.95hi"] = hpd.int.peak.titer[2]
                            
                            
                     }
                     
                     
                     
                     
                     
                     
                     
              }  
              
              
              
              # calculate RMSE and adjust times of infection         
              if(model == 1) {
                     
                     observed.data.for.fitting.single = observed.data.for.fitting
                     observed.data.for.fitting.single$time.since.infection = NA
                     observed.data.for.fitting.single$log.titer = observed.data.for.fitting.single$titer.pomona
                     
                     for(i in 1:N.inds){
                            idx.current = which(observed.data.for.fitting.single$id == i)
                            observed.data.for.fitting.single$time.since.infection[idx.current] = observed.data.for.fitting.single$time[idx.current] - post.individual[which(post.individual$model==model & post.individual$id == i),"toi.mean"]
                     }
                     
                     
                     post.overall[model,"RMSE"] = round(RMSE.fun.linear.individual.decay(peak.titers = post.individual[which(post.individual$model==model),"peak.titer.mean"],slopes = post.individual[which(post.individual$model==model),"decay.rate.mean"], observed.data = observed.data.for.fitting.single) ,5)
                     
              }
              
              if(model == 2) {
                     
                     
                     observed.data.for.fitting.double = observed.data.for.fitting
                     observed.data.for.fitting.double$time.since.infection = NA
                     observed.data.for.fitting.double$titer = observed.data.for.fitting.double$titer.pomona
                     
                     for(i in 1:N.inds){
                            idx.current = which(observed.data.for.fitting.double$id == i)
                            observed.data.for.fitting.double$time.since.infection[idx.current] = observed.data.for.fitting.double$time[idx.current] - post.individual[which(post.individual$model==model & post.individual$id == i),"toi.mean"]
                     }
                     
                     
                     post.overall[model,"RMSE"] = round(RMSE.fun.double.exp.individual.decay(peak.titers = post.individual[which(post.individual$model==model),"peak.titer.mean"],decay.rate = -post.individual[which(post.individual$model==model),"decay.rate.mean"], observed.data = observed.data.for.fitting.double) ,5)
                     
                     
              }
              
              if(model == 3) {
                     observed.data.for.fitting.power = observed.data.for.fitting
                     observed.data.for.fitting.power$time.since.infection = NA
                     observed.data.for.fitting.power$titer = observed.data.for.fitting.power$titer.pomona
                     
                     for(i in 1:N.inds){
                            idx.current = which(observed.data.for.fitting.power$id == i)
                            observed.data.for.fitting.power$time.since.infection[idx.current] = observed.data.for.fitting.power$time[idx.current] - post.individual[which(post.individual$model==model & post.individual$id == i),"toi.mean"]
                     }
                     
                     
                     post.overall[model,"RMSE"] = round(RMSE.fun.power.individual.decay(peak.titers = post.individual[which(post.individual$model==model),"peak.titer.mean"],shape = post.individual[which(post.individual$model==model),"shape.mean"],scale = post.individual[which(post.individual$model==model),"scale.mean"], observed.data = observed.data.for.fitting.power),5) 
              }
              
       }
       
       post.overall$gain[which(post.overall$model==1)] = mean(post.individual$toi.95.information.gained[which(post.individual$model==1)])
       post.overall$gain[which(post.overall$model==2)] = mean(post.individual$toi.95.information.gained[which(post.individual$model==2)])
       post.overall$gain[which(post.overall$model==3)] = mean(post.individual$toi.95.information.gained[which(post.individual$model==3)])
       
       
       post.overall$model[1] = "Single exp"
       post.overall$model[2] = "Double exp"
       post.overall$model[3] = "Power"
       
       post.individual$model[which(post.individual$model==1)] = "Single exp"
       post.individual$model[which(post.individual$model==2)] = "Double exp"
       post.individual$model[which(post.individual$model==3)] = "Power"
       
       post.overall$model = factor(post.overall$model,levels = c("Single exp","Double exp","Power"))
       post.individual$model = factor(post.individual$model,levels = c("Single exp","Double exp","Power"))
       
       all.fun.post.overall = post.overall
       
       kable(all.fun.post.overall ,"html") %>%
              kable_styling(bootstrap_options  =  c("striped", "hover","condensed"),full_width = F)
       
       # export posterior estimates
       saveRDS(all.fun.post.overall,"Function_comparison_post_overall.RDS")
}
rm(chains.burn.df.1,chains.burn.df.2,chains.burn.df.3, chains.burn.df.current)
gc()
##              used   (Mb) gc trigger    (Mb) limit (Mb)   max used   (Mb)
## Ncells    2879092  153.8    5322678   284.3         NA    4048362  216.3
## Vcells 1087145065 8294.3 1548713505 11815.8      32768 1290510572 9845.9

3 Output

Double exponential function only.

3.0.1 Stats

chains.burn.df.loglik = chains.burn.df[,grep("LogLik",colnames(chains.burn.df))]

chains.burn.df.loglik = chains.burn.df.loglik[is.finite(rowSums(chains.burn.df.loglik)),]

loglik.all.matrix = as.matrix(chains.burn.df.loglik)
rm(chains.burn.df.loglik)

LOOIC.val = loo(loglik.all.matrix)
LOOIC.val.est = LOOIC.val$estimates["looic",][1]

peak.titer.individual.means.pomona = apply(chains.burn.df[,paste0("peak.titer.pomona.",uid)],MARGIN = 2,mean)
decay.rate.individual.means.pomona = apply(chains.burn.df[,paste0("decay.rate.pomona.",uid)],MARGIN = 2,mean)

observed.data.for.fitting.temp = observed.data.for.fitting
colnames(observed.data.for.fitting.temp)[4] = "titer"
RMSE.val.pomona = round(RMSE.fun.double.exp.individual.decay(peak.titers = peak.titer.individual.means.pomona,decay.rates = -decay.rate.individual.means.pomona, observed.data = observed.data.for.fitting.temp),3)

peak.titer.individual.means.aut = apply(chains.burn.df[,paste0("peak.titer.aut.",uid)],MARGIN = 2,mean)
decay.rate.individual.means.aut = apply(chains.burn.df[,paste0("decay.rate.aut.",uid)],MARGIN = 2,mean)

observed.data.for.fitting.temp = observed.data.for.fitting
colnames(observed.data.for.fitting.temp)[4] = "titer"
RMSE.val.aut = round(RMSE.fun.double.exp.individual.decay(peak.titers = peak.titer.individual.means.aut,decay.rates = -decay.rate.individual.means.aut, observed.data = observed.data.for.fitting.temp),3)
LOOIC value: 5089.5856178
Root mean square error (pomona): 0.928.
Root mean square error (aut): 1.523.

3.0.2 Lab effect

CDC vs Ithaca

dens = density(chains.burn.df$lab.effect)
lab.effect.mean = mean(chains.burn.df$lab.effect)
hpd.int.lab.effect = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  

lab.fit = MASS::fitdistr(chains.burn.df$lab.effect,densfun = "normal")

Antibody levels for samples (Pomona or Autumnalis) not tested at CDC are on average 0.5929225 (95% CrI 0.3893709-0.7957079; SD = 0.103091) higher.

This is taken into account when fitting all model parameters.


plot.lab.effect.1 = ggplot(data=chains.burn.df,aes(x=lab.effect)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=lab.effect.mean,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept=hpd.int.lab.effect[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.lab.effect[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Lab effect") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )


plot.lab.effect.2 = plot.MCMC.chains(input.data = chains.burn.df.original, column.to.plot = which(colnames(chains.burn.df)=="lab.effect"),y.axis.label = "Lab effect",thinning=10)
plot.lab.effect.1

plot.lab.effect.2

3.0.3 Pomona

3.0.3.1 Posterior estimate mean peak titer:


dens = density(chains.burn.df$peak.titer.overall.pomona)
hpd.int.peak.titer.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
peak.titer.overall.maxdens.pomona=round(dens$x[which.max(dens$y)],2)
peak.titer.overall.mean.pomona = mean(chains.burn.df$peak.titer.overall.pomona)
peak.titer.overall.median.pomona = median(chains.burn.df$peak.titer.overall.pomona)


dens = density(chains.burn.df$peak.titer.sd.overall.pomona)
hpd.int.peak.titer.sd.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
peak.titer.sd.overall.maxdens.pomona=round(dens$x[which.max(dens$y)],2)
peak.titer.sd.overall.mean.pomona = mean(chains.burn.df$peak.titer.sd.overall.pomona)
peak.titer.sd.overall.median.pomona = median(chains.burn.df$peak.titer.sd.overall.pomona)


plot1 = ggplot(data=chains.burn.df,aes(x=peak.titer.overall.pomona)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=peak.titer.overall.maxdens.pomona,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept=hpd.int.peak.titer.pomona[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.peak.titer.pomona[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Peak antibody level mean (Pomona) \n(log2 dilution)") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )


plot2 = plot.MCMC.chains(input.data = chains.burn.df.original, column.to.plot = which(colnames(chains.burn.df)=="peak.titer.overall.pomona"),y.axis.label = "Peak antibody level mean (Pomona) \n(log2 dilution)",thinning=10)
plot1

plot2


  • Posterior estimate (max dens): 6.97
  • Posterior estimate (mean): 6.9728046
  • Posterior estimate (median): 6.9725206
  • Highest density credible interval = 6.7004855 - 7.2452111

3.0.3.2 Posterior estimate standard deviation of peak titer:

plot1 = ggplot(data=chains.burn.df,aes(x=peak.titer.sd.overall.pomona)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=peak.titer.sd.overall.maxdens.pomona,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept=hpd.int.peak.titer.sd.pomona[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.peak.titer.sd.pomona[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Peak antibody level SD (Pomona) \n(log2 dilution)") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )

plot2 = plot.MCMC.chains(input.data = chains.burn.df.original, column.to.plot = which(colnames(chains.burn.df)=="peak.titer.sd.overall.pomona"), y.axis.label = "Peak antibody level SD (Pomona) \n(log2 dilution)",thinning=10)
plot1

plot2


  • Posterior estimate (max dens): 1.83
  • Posterior estimate (mean): 1.8422147
  • Posterior estimate (median): 1.839451
  • Highest density credible interval = 1.6189053 - 2.0755833


3.0.3.3 Peak titer distribution using the posterior mean estimates of peak titer mean and SD

Including 200 random draws from the posteriors:

Nsamp = 200  
means = sample(chains.burn.df$peak.titer.overall.pomona,Nsamp,replace = F)
sds = sample(chains.burn.df$peak.titer.sd.overall.pomona,Nsamp,replace = F)


ggplot(data = data.frame(x = 0:15),aes(x=x)) +
       scale_x_continuous(breaks=0:15) +
       mapply(function(mean,sd){
              stat_function(fun = dnorm, colour = brewer.pal(11,"Spectral")[10],alpha=0.5,args = list(mean = mean,sd = sd),size=0.3)
       },
       mean = means,
       sd = sds
       ) +
       stat_function(fun = dnorm, colour = brewer.pal(11,"Spectral")[1],args = list(mean = peak.titer.overall.mean.pomona,sd = peak.titer.sd.overall.mean.pomona),size=1.5) +
       xlab("Peak antibody level (Pomona) \n(log2 dilution)") +
       ylab("Posterior distribution") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       )






3.0.3.4 Posterior estimate mean decay rate:


dens = density(chains.burn.df$decay.rate.overall.pomona)
hpd.int.decay.rate.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
decay.rate.overall.maxdens.pomona=round(dens$x[which.max(dens$y)],6)
decay.rate.overall.mean.pomona = mean(chains.burn.df$decay.rate.overall.pomona)
decay.rate.overall.median.pomona = median(chains.burn.df$decay.rate.overall.pomona)


dens = density(chains.burn.df$decay.rate.sd.overall.pomona)
hpd.int.decay.rate.sd.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
decay.rate.sd.overall.maxdens.pomona=round(dens$x[which.max(dens$y)],6)
decay.rate.sd.overall.mean.pomona = mean(chains.burn.df$decay.rate.sd.overall.pomona)
decay.rate.sd.overall.median.pomona = median(chains.burn.df$decay.rate.sd.overall.pomona)


plot1 = ggplot(data=chains.burn.df,aes(x=decay.rate.overall.pomona)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=decay.rate.overall.mean.pomona,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept=hpd.int.decay.rate.pomona[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.decay.rate.pomona[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Decay rate mean (Pomona) \n(1/day)") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )

plot2 = plot.MCMC.chains(input.data = chains.burn.df.original, column.to.plot = which(colnames(chains.burn.df)=="decay.rate.overall.pomona"), y.axis.label = "Decay rate mean (Pomona) \n(1/day)",thinning=10)
plot1

plot2


  • Posterior estimate (max dens): 8.63^{-4}
  • Posterior estimate (mean): 8.6188719^{-4}
  • Posterior estimate (median): 8.6100223^{-4}
  • Highest density credible interval = 7.8336119^{-4} - 9.4118982^{-4}

3.0.3.5 Posterior estimate standard deviation of decay rate:

plot1 = ggplot(data=chains.burn.df,aes(x=decay.rate.sd.overall.pomona)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=decay.rate.sd.overall.mean.pomona,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept=hpd.int.decay.rate.sd.pomona[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.decay.rate.sd.pomona[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Decay rate SD (Pomona) \n(1/day)") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )


plot2 = plot.MCMC.chains(input.data = chains.burn.df.original, column.to.plot = which(colnames(chains.burn.df)=="decay.rate.sd.overall.pomona"), y.axis.label = "Decay rate SD (Pomona) \n(1/day)",thinning=10)
plot1

plot2


  • Posterior estimate (max dens): 3.46^{-4}
  • Posterior estimate (mean): 3.4942643^{-4}
  • Posterior estimate (median): 3.4812674^{-4}
  • Highest density credible interval = 2.7771398^{-4} - 4.237336^{-4}


3.0.3.6 Decay rate distribution using the posterior mean estimates of peak titer mean and SD

Including 200 random draws from the posteriors:

Nsamp = 200  
means = sample(chains.burn.df$decay.rate.overall.pomona,Nsamp,replace = F)
sds = sample(chains.burn.df$decay.rate.sd.overall.pomona,Nsamp,replace = F)


ggplot(data = data.frame(x = seq(-0.0002,0.0019,0.00001)),aes(x=x)) +
       mapply(function(mean,sd){
              stat_function(fun = dnorm, colour = brewer.pal(11,"Spectral")[10],alpha=0.5,args = list(mean = mean,sd = sd),size=0.3)
       },
       mean = means,
       sd = sds
       ) +
       stat_function(fun = dnorm, colour = brewer.pal(11,"Spectral")[1],args = list(mean = decay.rate.overall.mean.pomona,sd = decay.rate.sd.overall.mean.pomona),size=1.5) +
       xlab("Decay rate (Pomona) \n(1/day)") +
       ylab("Posterior distribution") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
       )

3.0.3.7 Joint posterior peak titer vs decay means

Population level means:

joint.dens = chains.burn.df[,c("decay.rate.overall.pomona","peak.titer.overall.pomona")]
joint.dens$dens = get_density(chains.burn.df$decay.rate.overall.pomona,chains.burn.df$peak.titer.overall.pomona,n = 80)


joint.plot = ggplot(joint.dens,aes(x = decay.rate.overall.pomona, y = peak.titer.overall.pomona,color=dens)) +
       geom_point(alpha = 0.4,size = 0.5) +
       scale_colour_gradientn(colors = rev(brewer.pal(n = 6,"Spectral")),name="Density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 14),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")) +
       xlab("Decay rate posterior mean (Pomona) \n(1/day)") +
       ylab("Peak antibody level \n posterior mean (Pomona) \n(log2 dilution)")


joint.plot


Individual posterior means:

# 1200 posterior values per individual    
n.it.each = 200
ind.means = data.frame(id = rep(1:N.inds,each = n.it.each*6),
                       peak.titer.pomona = NA,
                       peak.titer.aut = NA,
                       decay.rate.pomona = NA,
                       decay.rate.aut = NA,
                       iteration = NA)
ind.means.2 = data.frame(id = 1:N.inds,
                         peak.titer.pomona = NA,
                         peak.titer.aut = NA,
                         decay.rate.pomona = NA,
                         decay.rate.aut = NA)


for(i in 1:N.inds){
       ind.means$peak.titer.pomona[which(ind.means$id == i)] = chains.burn.df[which(chains.burn.df$iteration == tail(chains.burn.df$iteration,n.it.each)),paste0("peak.titer.pomona.",i)]
       ind.means$peak.titer.aut[which(ind.means$id == i)] = chains.burn.df[which(chains.burn.df$iteration == tail(chains.burn.df$iteration,n.it.each)),paste0("peak.titer.aut.",i)]
       ind.means$decay.rate.pomona[which(ind.means$id == i)] = chains.burn.df[which(chains.burn.df$iteration == tail(chains.burn.df$iteration,n.it.each)),paste0("decay.rate.pomona.",i)]
       ind.means$decay.rate.aut[which(ind.means$id == i)] = chains.burn.df[which(chains.burn.df$iteration == tail(chains.burn.df$iteration,n.it.each)),paste0("decay.rate.aut.",i)]
       ind.means$iteration = chains.burn.df[which(chains.burn.df$iteration == tail(chains.burn.df$iteration,n.it.each)),"iteration"]
       
       
       ind.means.2$peak.titer.pomona[which(ind.means.2$id == i)] = mean(chains.burn.df[,paste0("peak.titer.pomona.",i)])
       ind.means.2$peak.titer.aut[which(ind.means.2$id == i)] = mean(chains.burn.df[,paste0("peak.titer.aut.",i)])
       ind.means.2$decay.rate.pomona[which(ind.means.2$id == i)] = mean(chains.burn.df[,paste0("decay.rate.pomona.",i)])
       ind.means.2$decay.rate.aut[which(ind.means.2$id == i)] = mean(chains.burn.df[,paste0("decay.rate.aut.",i)])
}





ind.means$dens = get_density(ind.means$peak.titer.pomona,ind.means$decay.rate.pomona,n = 80)



ggplot(ind.means,aes(x = decay.rate.pomona, y = peak.titer.pomona,color=dens)) +
       geom_point(alpha = 0.4,size = 0.5) +
       scale_colour_gradientn(colors = rev(brewer.pal(n = 6,"Spectral")),name="Density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             #legend.position = c(0.78,0.85),
             legend.text = element_text(size=10),
             legend.title = element_text(size=11),
             #legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       ) +
       xlab("Decay rate posterior samples \n(1/day)") +
       ylab("Peak antibody level posterior samples \n(log2 dilution)")

joint.plot.peak.decay = ggplot(ind.means,aes(x = decay.rate.pomona, y = peak.titer.pomona,color=dens)) +
       geom_point(alpha = 0.4,size = 0.5) +
       scale_colour_gradientn(colors = rev(brewer.pal(n = 6,"Spectral")),name="Density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             #legend.position = c(0.78,0.85),
             #legend.position = "none",
             legend.text = element_text(size=10),
             legend.title = element_text(size=11),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       ) +
       xlab("Decay rate posterior samples \n(1/day)") +
       ylab("Peak antibody level \n posterior samples \n(log2 dilution)") +
       labs(tag="(C)")

Correlation:

n.it = 400

posterior.cor = data.frame(iteration = rep(tail(unique(chains.burn.df$iteration),n.it),6),
                           chain = rep(1:6,each = n.it),
                           slope = NA,
                           p.val = NA,
                           adj.r2 = NA)
for(i in 1:nrow(posterior.cor)){
       cur.dat = data.frame(peak.titer = reshape2::melt(chains.burn.df[which(chains.burn.df$iteration==posterior.cor$iteration[i] & chains.burn.df$chain == posterior.cor$chain[i]),paste0("peak.titer.pomona.",1:N.inds)])[,2],
                            decay.rate = reshape2::melt(chains.burn.df[which(chains.burn.df$iteration==posterior.cor$iteration[i] & chains.burn.df$chain == posterior.cor$chain[i]),paste0("decay.rate.pomona.",1:N.inds)])[,2]
       )
       
       cur.lm = summary(lm(peak.titer~decay.rate,data = cur.dat))
       posterior.cor$slope[i] = cur.lm$coefficients[2]
       posterior.cor$p.val[i] = round(cur.lm$coefficients[8],5)
       posterior.cor$adj.r2[i] = cur.lm$adj.r.squared
}


dens = density(posterior.cor$p.val)
hpd.int.p.val.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  

dens = density(posterior.cor$slope)
hpd.int.slope.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  

dens = density(posterior.cor$adj.r2)
hpd.int.adj.r2.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  

Median p value: 0.52442 (95% CrI 0.0190309-1.0010681.
Proportion of P values below 0.05: 0.5180064

Mean slope: 89.5700803 (95% CrI -456.123899-646.1128699.

Median adjusted R^2: -0.0019445 (95% CrI -0.0043651-0.0085731.

ggplot(data=posterior.cor,aes(x=slope)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=hpd.int.slope.pomona[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.slope.pomona[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Slope decay rate vs peak antibody level") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )

ggplot(data=posterior.cor,aes(x=p.val)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=hpd.int.p.val.pomona[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.p.val.pomona[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("P value regression decay rate vs peak antibody level") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )

ggplot(data=posterior.cor,aes(x=adj.r2)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=mean(posterior.cor$adj.r2),col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept=hpd.int.adj.r2.pomona[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.adj.r2.pomona[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Adj. R^2 decay rate vs peak antibody level") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )

Using one posterior mean for each individual:

ggplot(ind.means.2,aes(x = decay.rate.pomona, y = peak.titer.pomona)) +
       geom_point(alpha = 1,size = 0.8,col = brewer.pal(n = 11,"Spectral")[10]) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 14),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")) +
       xlab("Decay rate posterior \nindividual means (Pomona) \n(1/day)") +
       ylab("Peak antibody level \nposterior individual means (Pomona) \n(log2 dilution)")

Correlation using posterior means:

lm1 = lm(peak.titer.pomona~decay.rate.pomona,data = ind.means.2)
# plot(lm1)
summary(lm1)
## 
## Call:
## lm(formula = peak.titer.pomona ~ decay.rate.pomona, data = ind.means.2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.2180 -1.0390  0.0694  1.0405  4.4549 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           8.1022     0.3239  25.016  < 2e-16 ***
## decay.rate.pomona -1310.5320   361.8998  -3.621 0.000343 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.529 on 305 degrees of freedom
## Multiple R-squared:  0.04122,    Adjusted R-squared:  0.03808 
## F-statistic: 13.11 on 1 and 305 DF,  p-value: 0.0003432

Two functions taken from the predicted values of the regression model:

exp.dat.1 = data.frame(x = 1:3000,y = exp.fun(start.titer = 7.5,rate = -0.0004,time = 1:3000))
exp.dat.2 = data.frame(x = 1:3000,y = exp.fun(start.titer = 6.5,rate = -0.0011,time = 1:3000))

plot(1:3000,exp.dat.1$y,type="l",ylim = c(0,8))
lines(1:3000,exp.dat.2$y,col="red")





3.0.3.8 Fitted function


(red = estimated function)

fit.dat = exp.fun(start.titer = peak.titer.overall.mean.pomona, rate = -decay.rate.overall.mean.pomona, time = 0:max(observed.data.for.fitting$time))

fit.dat = data.frame(titer = fit.dat, time = 0:max(observed.data.for.fitting$time))


ggplot() +
       geom_line(data = observed.data.for.fitting,aes(x = time,y = titer.pomona,col = id,group = id),alpha = 0.4,size = 0.6) +
       geom_point(data = observed.data.for.fitting,aes(x = time,y = titer.pomona,col = id,group = id),size=1.2,alpha = 0.9) +
       geom_line(data=fit.dat,aes(x = time, y = titer),col=brewer.pal(11,"Spectral")[1],size=1.3) +
       scale_color_gradientn(colours  =  brewer.pal(4,"Spectral")) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 12),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"))+
       xlab("Time since first positive sample (days)") +
       ylab("Antibody level (Pomona) \n(log2 dilution)")


Time = time since peak antibody level

fit.dat.pomona = exp.fun(start.titer = peak.titer.overall.mean.pomona, rate = -decay.rate.overall.mean.pomona, time = 0:max(observed.data.for.fitting$time.since.peak))

fit.dat.pomona = data.frame(titer = fit.dat.pomona, time = 0:max(observed.data.for.fitting$time.since.peak))


ggplot() +
       geom_line(data = observed.data.for.fitting,aes(x = time.since.peak,y = titer.pomona,col = id,group = id),alpha = 0.4,size = 0.6) +
       geom_point(data = observed.data.for.fitting,aes(x = time.since.peak,y = titer.pomona,col = id,group = id),size=1.2,alpha = 0.9) +
       geom_line(data=fit.dat.pomona,aes(x = time, y = titer),col=brewer.pal(11,"Spectral")[1],size=1.3) +
       scale_color_gradientn(colours  =  brewer.pal(4,"Spectral")) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 12),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"))+
       xlab("Time since peak antibody level (days)") +
       ylab("Antibody level (Pomona) \n(log2 dilution)")

Including fitted functions for 200 random samples from the posterior distribution (within 95% credible interval):

# sample N values from posterior distribution to get sense of error

N.samp = 200


set.seed(1234);sampled.iterations = sample(which(chains.burn.df$peak.titer.overall.pomona > hpd.int.peak.titer.pomona[1] & chains.burn.df$peak.titer.overall.pomona < hpd.int.peak.titer.pomona[2]), size = N.samp, replace = T)

peak.titer.samp.mean.pomona = chains.burn.df$peak.titer.overall.pomona[sampled.iterations]

peak.titer.samp.sd.pomona = chains.burn.df$peak.titer.sd.overall.pomona[sampled.iterations]


peak.titer.samp.df.pomona = data.frame(peak.titer.samp.mean.pomona,peak.titer.samp.sd.pomona)

peak.titer.samp.fun = function(x) rnorm(1,x[1],x[2])

peak.titer.samp.pomona = apply(peak.titer.samp.df.pomona,1,peak.titer.samp.fun)

decay.rate.samp.mean.pomona = chains.burn.df$decay.rate.overall.pomona[sampled.iterations]

decay.rate.samp.pomona = mean(chains.burn.df$decay.rate.overall.pomona)

plot.times = 0:max(observed.data.for.fitting$time.since.peak)

fit.dat.cloud.pomona = data.frame(time = rep(plot.times,N.samp), samp = rep(1:N.samp,each = length(plot.times)), titer = NA)

for(i in 1:N.samp){
       
       fit.dat.cloud.pomona$titer[which(fit.dat.cloud.pomona$samp == i)] = exp.fun(start.titer = peak.titer.samp.pomona[i], rate = -decay.rate.samp.pomona, time = plot.times)
       
}



ggplot() +
       geom_line(data=fit.dat.cloud.pomona, aes(x = time, y = titer, group = samp), col = brewer.pal(11,"Spectral")[10],alpha=0.4,size=0.3) +
       geom_line(data=fit.dat.pomona,aes(x = time, y = titer),col=brewer.pal(11,"Spectral")[1],size=1.5) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 12),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"))+
       xlab("Time since peak antibody level (days)") +
       ylab("Antibody level (Pomona) \n(log2 dilution)")

3.0.4 Autumnalis

3.0.4.1 Posterior estimate mean peak titer:


dens = density(chains.burn.df$peak.titer.overall.aut)
hpd.int.peak.titer.aut = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
peak.titer.overall.maxdens.aut=round(dens$x[which.max(dens$y)],2)
peak.titer.overall.mean.aut = mean(chains.burn.df$peak.titer.overall.aut)
peak.titer.overall.median.aut = median(chains.burn.df$peak.titer.overall.aut)


dens = density(chains.burn.df$peak.titer.sd.overall.aut)
hpd.int.peak.titer.sd.aut = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
peak.titer.sd.overall.maxdens.aut=round(dens$x[which.max(dens$y)],2)
peak.titer.sd.overall.mean.aut = mean(chains.burn.df$peak.titer.sd.overall.aut)
peak.titer.sd.overall.median.aut = median(chains.burn.df$peak.titer.sd.overall.aut)


plot1 = ggplot(data=chains.burn.df,aes(x=peak.titer.overall.aut)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=peak.titer.overall.maxdens.aut,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept=hpd.int.peak.titer.aut[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.peak.titer.aut[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Peak antibody level mean (Autumnalis) \n(log2 dilution)") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )

plot2 = plot.MCMC.chains(input.data = chains.burn.df, column.to.plot = which(colnames(chains.burn.df)=="peak.titer.overall.aut"),y.axis.label = "Peak antibody level mean (Autumnalis) \n(log2 dilution)",thinning=10)
plot1

plot2


  • Posterior estimate (max dens): 7.46
  • Posterior estimate (mean): 7.4632364
  • Posterior estimate (median): 7.4614985
  • Highest density credible interval = 7.1050021 - 7.8267388

3.0.4.2 Posterior estimate standard deviation of peak titer:

plot1 = ggplot(data=chains.burn.df,aes(x=peak.titer.sd.overall.aut)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=peak.titer.sd.overall.maxdens.aut,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept=hpd.int.peak.titer.sd.aut[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.peak.titer.sd.aut[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Peak antibody level SD (Autumnalis) \n(log2 dilution)") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )
plot2 = plot.MCMC.chains(input.data = chains.burn.df, column.to.plot = which(colnames(chains.burn.df)=="peak.titer.sd.overall.aut"), y.axis.label = "Peak antibody level SD (Autumnalis) \n(log2 dilution)",thinning=10)
plot1

plot2


  • Posterior estimate (max dens): 2.09
  • Posterior estimate (mean): 2.0982936
  • Posterior estimate (median): 2.0937291
  • Highest density credible interval = 1.8218543 - 2.3859862


3.0.4.3 Peak titer distribution using the posterior mean estimates of peak titer mean and SD

Including 200 random draws from the posteriors:

Nsamp = 200  
means = sample(chains.burn.df$peak.titer.overall.aut,Nsamp,replace = F)
sds = sample(chains.burn.df$peak.titer.sd.overall.aut,Nsamp,replace = F)


ggplot(data = data.frame(x = 0:15),aes(x=x)) +
       scale_x_continuous(breaks=0:15) +
       mapply(function(mean,sd){
              stat_function(fun = dnorm, colour = brewer.pal(11,"Spectral")[10],alpha=0.5,args = list(mean = mean,sd = sd),size=0.3)
       },
       mean = means,
       sd = sds
       ) +
       stat_function(fun = dnorm, colour = brewer.pal(11,"Spectral")[1],args = list(mean = peak.titer.overall.mean.aut,sd = peak.titer.sd.overall.mean.aut),size=1.5) +
       xlab("Peak antibody level (Autumnalis) \n(log2 dilution)") +
       ylab("Posterior distribution") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       )






3.0.4.4 Posterior estimate mean decay rate:


dens = density(chains.burn.df$decay.rate.overall.aut)
hpd.int.decay.rate.aut = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
decay.rate.overall.maxdens.aut=round(dens$x[which.max(dens$y)],6)
decay.rate.overall.mean.aut = mean(chains.burn.df$decay.rate.overall.aut)
decay.rate.overall.median.aut = median(chains.burn.df$decay.rate.overall.aut)


dens = density(chains.burn.df$decay.rate.sd.overall.aut)
hpd.int.decay.rate.sd.aut = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
decay.rate.sd.overall.maxdens.aut=round(dens$x[which.max(dens$y)],6)
decay.rate.sd.overall.mean.aut = mean(chains.burn.df$decay.rate.sd.overall.aut)
decay.rate.sd.overall.median.aut = median(chains.burn.df$decay.rate.sd.overall.aut)


plot1 = ggplot(data=chains.burn.df,aes(x=decay.rate.overall.aut)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=decay.rate.overall.mean.aut,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept=hpd.int.decay.rate.aut[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.decay.rate.aut[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Decay rate mean (Autumnalis) \n(1/day)") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )
plot2 = plot.MCMC.chains(input.data = chains.burn.df, column.to.plot = which(colnames(chains.burn.df)=="decay.rate.overall.aut"), y.axis.label = "Decay rate mean (Autumnalis) \n(1/day)",thinning=10)
plot1

plot2


  • Posterior estimate (max dens): 6.57^{-4}
  • Posterior estimate (mean): 6.5922361^{-4}
  • Posterior estimate (median): 6.5793416^{-4}
  • Highest density credible interval = 5.6762402^{-4} - 7.5425992^{-4}

3.0.4.5 Posterior estimate standard deviation of decay rate:

plot1 = ggplot(data=chains.burn.df,aes(x=decay.rate.sd.overall.aut)) +
       geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
       geom_vline(xintercept=decay.rate.sd.overall.mean.aut,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept=hpd.int.decay.rate.sd.aut[1],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept=hpd.int.decay.rate.sd.aut[2],col="black",size=0.8,linetype="dotted",alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       xlab("Decay rate SD (Autumnalis) \n(1/day)") +
       ylab("Posterior density") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = "none"
       )


plot2 = plot.MCMC.chains(input.data = chains.burn.df, column.to.plot = which(colnames(chains.burn.df)=="decay.rate.sd.overall.aut"), y.axis.label = "Decay rate SD (Autumnalis) \n(1/day)",thinning=10)
plot1

plot2


  • Posterior estimate (max dens): 2.72^{-4}
  • Posterior estimate (mean): 2.7481138^{-4}
  • Posterior estimate (median): 2.7319565^{-4}
  • Highest density credible interval = 1.8929749^{-4} - 3.6438391^{-4}


3.0.4.6 Decay rate distribution using the posterior mean estimates of peak titer mean and SD

Including 200 random draws from the posteriors:

Nsamp = 200  
means = sample(chains.burn.df$decay.rate.overall.aut,Nsamp,replace = F)
sds = sample(chains.burn.df$decay.rate.sd.overall.aut,Nsamp,replace = F)


ggplot(data = data.frame(x = seq(-0.0002,0.0016,0.00001)),aes(x=x)) +
       mapply(function(mean,sd){
              stat_function(fun = dnorm, colour = brewer.pal(11,"Spectral")[10],alpha=0.5,args = list(mean = mean,sd = sd),size=0.3)
       },
       mean = means,
       sd = sds
       ) +
       stat_function(fun = dnorm, colour = brewer.pal(11,"Spectral")[1],args = list(mean = decay.rate.overall.mean.aut,sd = decay.rate.sd.overall.mean.aut),size=1.5) +
       xlab("Decay rate (Autumnalis) \n(1/day)") +
       ylab("Posterior distribution") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
       )

3.0.4.7 Joint posterior peak titer vs decay means

joint.dens = chains.burn.df[,c("decay.rate.overall.aut","peak.titer.overall.aut")]
joint.dens$dens = get_density(chains.burn.df$decay.rate.overall.aut,chains.burn.df$peak.titer.overall.aut,n = 80)


joint.plot = ggplot(joint.dens,aes(x = decay.rate.overall.aut, y = peak.titer.overall.aut,color=dens)) +
       geom_point(alpha = 0.4,size = 0.5) +
       scale_colour_gradientn(colors = rev(brewer.pal(n = 6,"Spectral")),name="Density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 14),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")) +
       xlab("Decay rate posterior mean (Autumnalis) \n(1/day)") +
       ylab("Peak titer posterior mean (Autumnalis) \n(log2 dilution)")


joint.plot


Individual posterior means:

ind.means$dens = get_density(ind.means$peak.titer.aut,ind.means$decay.rate.aut,n = 80)


joint.plot = ggplot(ind.means,aes(x = decay.rate.aut, y = peak.titer.aut,color=dens)) +
       geom_point(alpha = 0.4,size = 0.5) +
       scale_colour_gradientn(colors = rev(brewer.pal(n = 6,"Spectral")),name="Density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 14),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")) +
       xlab("Decay rate posterior individual means (Autumnalis) \n(1/day)") +
       ylab("Peak antibody level \n posterior individual means (Autumnalis) \n(log2 dilution)")


joint.plot

ggplot(ind.means.2,aes(x = decay.rate.aut, y = peak.titer.aut)) +
       geom_point(alpha = 1,size = 0.8,col = brewer.pal(n = 11,"Spectral")[10]) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 14),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")) +
       xlab("Decay rate posterior individual means (Autumnalis) \n(1/day)") +
       ylab("Peak antibody level \n posterior individual means (Autumnalis) \n(log2 dilution)")

(points on one line = no Autumnalis sample available, so sampling from the prior.
variation along peak titer axis is due to the correlation between Pomona and Autumnalis peak level)





3.0.4.8 Fitted function


(red = estimated function)

fit.dat.aut = exp.fun(start.titer = peak.titer.overall.mean.aut, rate = -decay.rate.overall.mean.aut, time = 0:max(observed.data.for.fitting$time))

fit.dat.aut = data.frame(titer = fit.dat.aut, time = 0:max(observed.data.for.fitting$time))


ggplot() +
       geom_line(data = observed.data.for.fitting,aes(x = time,y = titer.aut,col = id,group = id),alpha = 0.4,size = 0.6) +
       geom_point(data = observed.data.for.fitting,aes(x = time,y = titer.aut,col = id,group = id),size=1.2,alpha = 0.9) +
       geom_line(data=fit.dat.aut,aes(x = time, y = titer),col=brewer.pal(11,"Spectral")[1],size=1.3) +
       scale_color_gradientn(colours  =  brewer.pal(4,"Spectral")) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 12),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"))+
       xlab("Time since first positive sample (days)") +
       ylab("Antibody level (Autumnalis) \n(log2 dilution)")


Time = time since peak antibody level

fit.dat.aut = exp.fun(start.titer = peak.titer.overall.mean.aut, rate = -decay.rate.overall.mean.aut, time = 0:max(observed.data.for.fitting$time.since.peak))

fit.dat.aut = data.frame(titer = fit.dat.aut, time = 0:max(observed.data.for.fitting$time.since.peak))


ggplot() +
       geom_line(data = observed.data.for.fitting,aes(x = time.since.peak,y = titer.aut,col = id,group = id),alpha = 0.4,size = 0.6) +
       geom_point(data = observed.data.for.fitting,aes(x = time.since.peak,y = titer.aut,col = id,group = id),size=1.2,alpha = 0.9) +
       geom_line(data=fit.dat.aut,aes(x = time, y = titer),col=brewer.pal(11,"Spectral")[1],size=1.3) +
       scale_color_gradientn(colours  =  brewer.pal(4,"Spectral")) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 12),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"))+
       xlab("Time since peak antibody level (days)") +
       ylab("Antibody level (Autumnalis) \n(log2 dilution)")

Including fitted functions for 200 random samples from the posterior distribution (within 95% credible interval):

# sample N values from posterior distribution to get sense of error

N.samp = 200


set.seed(1234);sampled.iterations = sample(which(chains.burn.df$peak.titer.overall.aut > hpd.int.peak.titer.aut[1] & chains.burn.df$peak.titer.overall.aut < hpd.int.peak.titer.aut[2]), size = N.samp, replace = T)

peak.titer.samp.mean.aut = chains.burn.df$peak.titer.overall.aut[sampled.iterations]

peak.titer.samp.sd.aut = chains.burn.df$peak.titer.sd.overall.aut[sampled.iterations]


peak.titer.samp.df.aut = data.frame(peak.titer.samp.mean.aut,peak.titer.samp.sd.aut)

peak.titer.samp.fun = function(x) rnorm(1,x[1],x[2])

peak.titer.samp.aut = apply(peak.titer.samp.df.aut,1,peak.titer.samp.fun)

decay.rate.samp.mean.aut = chains.burn.df$decay.rate.overall.aut[sampled.iterations]

decay.rate.samp.aut = mean(chains.burn.df$decay.rate.overall.aut)

plot.times = 0:max(observed.data.for.fitting$time.since.peak)

fit.dat.cloud.aut = data.frame(time = rep(plot.times,N.samp), samp = rep(1:N.samp,each = length(plot.times)), titer = NA)

for(i in 1:N.samp){
       
       fit.dat.cloud.aut$titer[which(fit.dat.cloud.aut$samp == i)] = exp.fun(start.titer = peak.titer.samp.aut[i], rate = -decay.rate.samp.aut, time = plot.times)
       
}



ggplot() +
       geom_line(data=fit.dat.cloud.aut, aes(x = time, y = titer, group = samp), col = brewer.pal(11,"Spectral")[10],alpha=0.4,size=0.3) +
       geom_line(data=fit.dat.aut,aes(x = time, y = titer),col=brewer.pal(11,"Spectral")[1],size=1.5) +
       scale_color_gradientn(colours  =  brewer.pal(4,"Spectral")) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 12),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"))+
       xlab("Time since peak antibody level (days)") +
       ylab("Antibody level (Autumnalis) \n(log2 dilution)")

3.1 Pomona and Autumnalis

3.1.1 Peak antibody level

Correlation between Pomona and Autumnalis peak antibody levels

r2.fun = function(x) {
       summary(lm(as.numeric(x[1:(length(x)/2)])~as.numeric(x[((length(x)/2)+1):length(x)])))$r.squared
}

id.aut.not.all.na = unique(observed.data.for.fitting$id[which(!is.na(observed.data.for.fitting$titer.aut))])

r2s = apply(chains.burn.df[which(chains.burn.df$iteration %in% tail(chains.burn.df$iteration,1000)),c(paste0("peak.titer.aut.",id.aut.not.all.na),paste0("peak.titer.pomona.",id.aut.not.all.na))],1,r2.fun)

dens = density(r2s)
hpd.int.r2s = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
maxdens.r2s=round(dens$x[which.max(dens$y)],2)



intercept.fun = function(x) {
       summary(lm(as.numeric(x[1:(length(x)/2)])~as.numeric(x[((length(x)/2)+1):length(x)])))$coefficients[1]
}

intercepts = apply(chains.burn.df[which(chains.burn.df$iteration %in% tail(chains.burn.df$iteration,1000)),c(paste0("peak.titer.aut.",id.aut.not.all.na),paste0("peak.titer.pomona.",id.aut.not.all.na))],1,intercept.fun)

dens = density(intercepts)
hpd.int.intercepts = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
maxdens.intercepts=round(dens$x[which.max(dens$y)],2)


slope.fun = function(x) {
       summary(lm(as.numeric(x[1:(length(x)/2)])~as.numeric(x[((length(x)/2)+1):length(x)])))$coefficients[2]
}

slopes = apply(chains.burn.df[which(chains.burn.df$iteration %in% tail(chains.burn.df$iteration,1000)),c(paste0("peak.titer.aut.",id.aut.not.all.na),paste0("peak.titer.pomona.",id.aut.not.all.na))],1,slope.fun)

dens = density(slopes)
hpd.int.slopes = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)  
maxdens.slopes=round(dens$x[which.max(dens$y)],2)

R^2 = 0.98, 95% CrI = 0.8702386 - 1.0021471.
Intercept = -0.17, 95% CrI = -1.3400768 - 0.7453464.
(= Aut peak level - Pom peak level)
Effect estimate = 1.11, 95% CrI = 0.9746661 - 1.25398.
(= slope)

Nsamp = 200
means.pomona = sample(chains.burn.df$peak.titer.overall.pomona,Nsamp,replace = F)
sds.pomona = sample(chains.burn.df$peak.titer.sd.overall.pomona,Nsamp,replace = F)
means.aut = sample(chains.burn.df$peak.titer.overall.aut,Nsamp,replace = F)
sds.aut = sample(chains.burn.df$peak.titer.sd.overall.aut,Nsamp,replace = F)


ggplot(data = data.frame(x = 0:15),aes(x=x)) +
       scale_x_continuous(breaks=0:15) +
       
       mapply(function(mean,sd){
              stat_function(fun = dnorm, args = list(mean = mean,sd = sd),aes(color="Autumnalis"),size=0.4)
       },
       mean = means.aut,
       sd = sds.aut
       ) +
       stat_function(fun = dnorm, args = list(mean = peak.titer.overall.mean.aut,sd = peak.titer.sd.overall.mean.aut),color =  brewer.pal(11,"Spectral")[1],size=1.5) +
       geom_vline(xintercept=peak.titer.overall.mean.aut,col=brewer.pal(11,"Spectral")[1],size=0.8,alpha=0.6) +
       mapply(function(mean,sd){
              stat_function(fun = dnorm,args = list(mean = mean,sd = sd),aes(color="Pomona"),size=0.4)
       },
       mean = means.pomona,
       sd = sds.pomona
       ) +
       stat_function(fun = dnorm, args = list(mean = peak.titer.overall.mean.pomona,sd = peak.titer.sd.overall.mean.pomona),color =  brewer.pal(11,"Spectral")[10],size=1.5) +
       geom_vline(xintercept=peak.titer.overall.mean.pomona,col=brewer.pal(11,"Spectral")[10],size=0.8,alpha=0.6) +
       scale_color_manual("Serovar",values = alpha(c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1]),0.3)) +
       xlab("Peak antibody level \n(log2 dilution)") +
       ylab("Density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = c(0.85,0.75),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent")
       )

plot1 = ggplot(data = data.frame(x = 0:15),aes(x=x)) +
       scale_x_continuous(breaks=seq(0,15,2),limits = c(0,16)) +
       mapply(function(mean,sd){
              stat_function(fun = dnorm, args = list(mean = mean,sd = sd),aes(color="Autumnalis"),size=0.2,alpha=0.1)
       },
       mean = means.aut,
       sd = sds.aut
       ) +
       mapply(function(mean,sd){
              stat_function(fun = dnorm,args = list(mean = mean,sd = sd),aes(color="Pomona"),size=0.2,alpha=0.1)
       },
       mean = means.pomona,
       sd = sds.pomona
       ) +
       stat_function(fun = dnorm, args = list(mean = peak.titer.overall.mean.aut,sd = peak.titer.sd.overall.mean.aut),color =  brewer.pal(11,"Spectral")[1],size=1.5) +
       stat_function(fun = dnorm, args = list(mean = peak.titer.overall.mean.pomona,sd = peak.titer.sd.overall.mean.pomona),color =  brewer.pal(11,"Spectral")[10],size=1.5) +
       #geom_vline(xintercept=peak.titer.overall.mean.aut,col=brewer.pal(11,"Spectral")[1],size=1,alpha=1) +
       #geom_vline(xintercept=peak.titer.overall.mean.pomona,col=brewer.pal(11,"Spectral")[10],size=1,alpha=1) +
       scale_color_manual("Serovar",values = alpha(c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1]),0.3)) +
       xlab("Peak antibody level \n(log2 dilution)") +
       ylab("Density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.title = element_text(size = 10),
             legend.text = element_text(size = 9),
             #legend.position = c(0.78,0.85),
             legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       ) +
       labs(tag = "(A)")

Individual level means:

cor.plot.individual = ggplot(ind.means.2,aes(x = peak.titer.pomona, y = peak.titer.aut)) +
       geom_point(alpha = 1,size = 0.8,col = brewer.pal(n = 11,"Spectral")[10]) +
       theme_light(base_family = "Avenir Next") +
       scale_x_continuous(breaks = seq(0,15,2)) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme(plot.title  =  element_text(size  = 13),
             text = element_text(size = 13),
             axis.text = element_text(size = 13),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")) +
       xlab("Peak antibody level (log2 dilution) \n posterior individual means \n (Pomona)") +
       ylab("Peak antibody level (log2 dilution) \n posterior individual means \n (Autumnalis)") +
       labs(tag = "(A)")

cor.plot.individual

joint.dens = chains.burn.df[,c("peak.titer.overall.pomona","peak.titer.overall.aut")]
joint.dens$dens = get_density(chains.burn.df$peak.titer.overall.pomona,chains.burn.df$peak.titer.overall.aut,n = 80)


joint.plot.peak.correlation = ggplot(joint.dens,aes(x = peak.titer.overall.pomona, y = peak.titer.overall.aut,color=dens)) +
       geom_point(alpha = 0.4,size = 0.5) +
       scale_colour_gradientn(colors = rev(brewer.pal(n = 6,"Spectral")),name="Density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 13),
             text = element_text(size = 13),
             axis.text = element_text(size = 13),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")) +
       xlab("Peak antibody level (log2 dilution) \n  posterior overall mean \n (Pomona)") +
       ylab("Peak antibody level (log2 dilution) \n  posterior overall mean \n (Autumnalis)") +
       labs(tag = "(B)")


joint.plot.peak.correlation

Combined plot peak titer correlations for manuscript

comb.plot.correlation = cor.plot.individual + joint.plot.peak.correlation


comb.plot.correlation

3.1.2 Decay rate

Nsamp = 200
means.pomona = sample(chains.burn.df$decay.rate.overall.pomona,Nsamp,replace = F)
sds.pomona = sample(chains.burn.df$decay.rate.sd.overall.pomona,Nsamp,replace = F)
means.aut = sample(chains.burn.df$decay.rate.overall.aut,Nsamp,replace = F)
sds.aut = sample(chains.burn.df$decay.rate.sd.overall.aut,Nsamp,replace = F)


ggplot(data = data.frame(x = seq(0,0.0016,0.0001)),aes(x=x)) +
       #scale_x_continuous(breaks=0:15) +
       
       mapply(function(mean,sd){
              stat_function(fun = dnorm, args = list(mean = mean,sd = sd),aes(color="Autumnalis"),size=0.4)
       },
       mean = means.aut,
       sd = sds.aut
       ) +
       stat_function(fun = dnorm, args = list(mean = decay.rate.overall.mean.aut,sd = decay.rate.sd.overall.mean.aut),color =  brewer.pal(11,"Spectral")[1],size=1.5) +
       geom_vline(xintercept=decay.rate.overall.mean.aut,col=brewer.pal(11,"Spectral")[1],size=0.8,alpha=0.6) +
       mapply(function(mean,sd){
              stat_function(fun = dnorm,args = list(mean = mean,sd = sd),aes(color="Pomona"),size=0.4)
       },
       mean = means.pomona,
       sd = sds.pomona
       ) +
       stat_function(fun = dnorm, args = list(mean = decay.rate.overall.mean.pomona,sd = decay.rate.sd.overall.mean.pomona),color =  brewer.pal(11,"Spectral")[10],size=1.5) +
       geom_vline(xintercept=decay.rate.overall.mean.pomona,col=brewer.pal(11,"Spectral")[10],size=0.8,alpha=0.6) +
       scale_color_manual("Serovar",values = alpha(c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1]),0.3)) +
       xlab("Decay rate (1/day)") +
       ylab("Density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = c(0.85,0.75),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent")
       )

plot2 = ggplot(data = data.frame(x = seq(0,0.0017,0.0001)),aes(x=x)) +
       #scale_x_continuous(breaks=0:15) +
       
       mapply(function(mean,sd){
              stat_function(fun = dnorm, args = list(mean = mean,sd = sd),aes(color="Autumnalis"),size=0.2,alpha=0.1)
       },
       mean = means.aut,
       sd = sds.aut
       ) +
       
       #geom_vline(xintercept=decay.rate.overall.mean.aut,col=brewer.pal(11,"Spectral")[1],size=0.8,alpha=0.6) +
       mapply(function(mean,sd){
              stat_function(fun = dnorm,args = list(mean = mean,sd = sd),aes(color="Pomona"),size=0.2,alpha=0.1)
       },
       mean = means.pomona,
       sd = sds.pomona
       ) +
       stat_function(fun = dnorm, args = list(mean = decay.rate.overall.mean.aut,sd = decay.rate.sd.overall.mean.aut),color =  brewer.pal(11,"Spectral")[1],size=1.5) +
       stat_function(fun = dnorm, args = list(mean = decay.rate.overall.mean.pomona,sd = decay.rate.sd.overall.mean.pomona),color =  brewer.pal(11,"Spectral")[10],size=1.5) +
       #geom_vline(xintercept=decay.rate.overall.mean.pomona,col=brewer.pal(11,"Spectral")[10],size=0.8,alpha=0.6) +
       scale_color_manual("Serovar",values = alpha(c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1]),0.3)) +
       scale_x_continuous(breaks = c(0,0.00075,0.0015),labels = c("0","0.00075","0.0015")) +
       xlab("Decay rate \n(1/day)") +
       ylab("Density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.title = element_text(size = 12),
             legend.text = element_text(size = 11),
             #legend.position = "none",
             legend.position = c(0.75,0.85),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       ) +
       labs(tag = "(B)")

3.1.3 Fitted function

ggplot() +
       #geom_line(data=fit.dat.cloud.aut, aes(x = time, y = titer, group = samp), col = brewer.pal(11,"Spectral")[1],alpha=0.5,size=0.3) +
       geom_line(data=fit.dat.aut,aes(x = time, y = titer,col="Autumnalis"),size=1.5) +
       #geom_line(data=fit.dat.cloud.pomona, aes(x = time, y = titer, group = samp), col = brewer.pal(11,"Spectral")[10],alpha=0.5,size=0.3) +
       geom_line(data=fit.dat.pomona,aes(x = time, y = titer,col="Pomona"),size=1.5) +
       scale_color_manual("Serovar",values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 12),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             #legend.position  =  "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"))+
       xlab("Time since peak antibody level (days)") +
       ylab("Antibody level (log2 dilution)")

ggplot() +
       geom_line(data=fit.dat.cloud.aut, aes(x = time, y = titer, group = samp, col = "Autumnalis"),alpha=0.5,size=0.3) +
       #geom_line(data=fit.dat.aut,aes(x = time, y = titer,col="Autumnalis"),size=1.5) +
       geom_line(data=fit.dat.cloud.pomona, aes(x = time, y = titer, group = samp, col = "Pomona"),alpha=0.5,size=0.3) +
       #geom_line(data=fit.dat.pomona,aes(x = time, y = titer,col="Pomona"),size=1.5) +
       scale_color_manual("Serovar",values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 12),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             #legend.position  =  "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"))+
       xlab("Time since peak antibody level (days)") +
       ylab("Antibody level (log2 dilution)")

plot3 = ggplot() +
       geom_line(data=fit.dat.cloud.aut, aes(x = time, y = titer, group = samp, col = "Autumnalis"),alpha=0.1,size=0.2) +
       geom_line(data=fit.dat.cloud.pomona, aes(x = time, y = titer, group = samp, col = "Pomona"),alpha=0.1,size=0.2) +
       geom_line(data=fit.dat.aut,aes(x = time, y = titer,col="Autumnalis"),size=1.5) +
       geom_line(data=fit.dat.pomona,aes(x = time, y = titer,col="Pomona"),size=1.5) +
       scale_color_manual("Serovar",values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title  =  element_text(size  = 12),
             text = element_text(size = 11),
             axis.text = element_text(size = 11),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)),
             legend.position  =  c(0.85,0.75),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"))+
       xlab("Time since peak antibody level (days)") +
       ylab("Antibody level (log2 dilution)")
labs(tag = "(C)")
## $tag
## [1] "(C)"
## 
## attr(,"class")
## [1] "labels"

3.2 Combined figures peak antibody level, decay rate, fitted functions

plot1 + plot2 + plot3 + 
       plot_layout(widths = c(1,1,2))

plot1 + plot2

joint.plot.peak.decay

comb.plot = plot1 + plot2 + joint.plot.peak.decay +
       plot_layout(widths = c(4,4,4))


comb.plot

3.3 Combined figure function comparison and fitted Pom and Aut

# import posterior estimates for the three functions

all.fun.post.overall = readRDS("Function_comparison_post_overall.RDS")

fit.dat.single = linear.fun(peak.titer = all.fun.post.overall$peak.titer.mean[1], slope = -all.fun.post.overall$decay.mean[1], time = 0:4000)
fit.dat.single = data.frame(titer = fit.dat.single, time = 0:4000)
fit.dat.single$model = "Single exp"
fit.dat.single = fit.dat.single[-which(fit.dat.single$titer<0),]


fit.dat.double = exp.fun(start.titer = all.fun.post.overall$peak.titer.mean[2], rate = -all.fun.post.overall$decay.mean[2], time = 0:4000)

fit.dat.double = data.frame(titer = fit.dat.double, time = 0:4000)

fit.dat.double$model = "Double exp"


fit.dat.double.aut = exp.fun(start.titer = all.fun.post.overall$peak.titer.aut.mean[2], rate = -all.fun.post.overall$decay.aut.mean[2], time = 0:4000)

fit.dat.double.aut = data.frame(titer = fit.dat.double.aut, time = 0:4000)

fit.dat.double.aut$model = "Double exp (Autumnalis)"



fit.dat.power = power.fun(peak.titer = all.fun.post.overall$peak.titer.mean[3], shape = all.fun.post.overall$shape.mean[3],scale = all.fun.post.overall$scale.mean[3], time = 0:4000)

fit.dat.power = data.frame(titer = fit.dat.power, time = 0:4000)

fit.dat.power$model = "Power"


fit.dat = rbind(fit.dat.single,fit.dat.double,fit.dat.power)

fit.dat$model = factor(fit.dat$model,levels = c("Single exp","Double exp","Power"))


plot.1.functions = ggplot(data=fit.dat,aes(x = time, y = titer,color=model,group=model)) +
       geom_line(size=1.2) +
       scale_color_manual("Model",values  =  brewer.pal(4,"Spectral"),labels = c("Single exponential","Double exponential","Power")) +
       scale_y_continuous(limits = c(0,8),breaks = 0:8) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = c(0.65,0.75),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(colour = "transparent", fill = "transparent")
       ) +
       xlab("Time since peak \nantibody level (days)") +
       ylab("Antibody level (log2 dilution)") +
       labs(tag="(A)")



plot2.with.aut = ggplot() +
       #geom_line(data = observed.data.for.fitting,aes(x = time.since.peak,y = titer.pomona,col = id,group = id),alpha = 0.2,size = 0.2,color=brewer.pal(11,"Spectral")[10]) +
       geom_point(data = observed.data.for.fitting,aes(x = time.since.peak,y = titer.pomona-0.1,col = id,group = id),size=1,alpha = 0.4,color=brewer.pal(11,"Spectral")[10]) +
       #geom_line(data = observed.data.for.fitting,aes(x = time.since.peak,y = titer.aut,col = id,group = id),alpha = 0.2,size = 0.2,color=brewer.pal(11,"Spectral")[1]) +
       geom_point(data = observed.data.for.fitting,aes(x = time.since.peak,y = titer.aut+0.1,col = id,group = id),size=1,alpha = 0.4,color=brewer.pal(11,"Spectral")[1]) +
       geom_line(data=fit.dat.double,aes(x = time, y = titer,col="Pomona"),size=1.6) +
       geom_line(data=fit.dat.double.aut,aes(x = time, y = titer,col="Autumnalis"),size=1.6) +
       #scale_color_gradientn(colours  =  brewer.pal(4,"Spectral")) +
       scale_color_manual("Serovar",values = alpha(c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1]),1)) +
       scale_y_continuous(limits = c(0,12),breaks = 0:12) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = c(0.7,0.8),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(colour = "transparent", fill = "transparent")
       ) +
       #ggtitle("Double exponential") +
       xlab("Time since peak \nantibody level (days)") +
       ylab("Antibody level (log2 dilution)") +
       labs(tag = "(B)")


plotA = plot.1.functions + plot2.with.aut
plotA

# dataframe to store estimated parameters

model.outputs = data.frame(id = 1:N.inds,
                           pittag = unique(observed.data.for.fitting$Pittag),
                           toi.mid.interval = round(neg.intervals[1:N.inds]/2),
                           toi.estimated.mean = NA,
                           toi.estimated.median = NA,
                           toi.estimated.maxdens = NA,
                           toi.estimated.mean.95ci.low = NA,
                           toi.estimated.mean.95ci.high = NA,
                           toi.estimated.maxdens.95hdi.low = NA,
                           toi.estimated.maxdens.95hdi.high = NA,
                           toi.information.gained.95cri.perc = NA,
                           toi.information.gained.80cri.perc = NA,
                           toi.information.gained.50cri.perc = NA,
                           kl.divergence = NA,
                           peak.titer.estimated.mean.pomona = NA,
                           peak.titer.estimated.median.pomona = NA,
                           peak.titer.estimated.maxdens.pomona = NA,
                           peak.titer.estimated.mean.95ci.low.pomona = NA,
                           peak.titer.estimated.mean.95ci.high.pomona = NA,
                           peak.titer.estimated.maxdens.95hdi.low.pomona = NA,
                           peak.titer.estimated.maxdens.95hdi.high.pomona = NA,
                           decay.rate.estimated.mean.pomona = NA,
                           decay.rate.estimated.median.pomona = NA,
                           decay.rate.estimated.maxdens.pomona = NA,
                           decay.rate.estimated.mean.95ci.low.pomona = NA,
                           decay.rate.estimated.mean.95ci.high.pomona = NA,
                           decay.rate.estimated.maxdens.95hdi.low.pomona = NA,
                           decay.rate.estimated.maxdens.95hdi.high.pomona = NA,
                           peak.titer.estimated.mean.aut = NA,
                           peak.titer.estimated.median.aut = NA,
                           peak.titer.estimated.maxdens.aut = NA,
                           peak.titer.estimated.mean.95ci.low.aut = NA,
                           peak.titer.estimated.mean.95ci.high.aut = NA,
                           peak.titer.estimated.maxdens.95hdi.low.aut = NA,
                           peak.titer.estimated.maxdens.95hdi.high.aut = NA,
                           decay.rate.estimated.mean.aut = NA,
                           decay.rate.estimated.median.aut = NA,
                           decay.rate.estimated.maxdens.aut = NA,
                           decay.rate.estimated.mean.95ci.low.aut = NA,
                           decay.rate.estimated.mean.95ci.high.aut = NA,
                           decay.rate.estimated.maxdens.95hdi.low.aut = NA,
                           decay.rate.estimated.maxdens.95hdi.high.aut = NA
)




# matrix for plotting individual posteriors of time of infection
# nrow = number of iterations to store
theta.all = matrix(data = NA, ncol = N.inds, nrow = 6000)  



# print figures?  

plot.figs = F

teller = 1
for(i in 1:N.inds){
       
       
       
       chains.current.individual.df = chains.burn.df[,c(paste0("peak.titer.pomona.",i),paste0("peak.titer.aut.",i),paste0("decay.rate.pomona.",i),paste0("decay.rate.aut.",i),paste0("toi.",i),"iteration","chain")]
       
       colnames(chains.current.individual.df) = c("peak.titer.pomona","peak.titer.aut","decay.rate.pomona","decay.rate.aut","toi","iteration","chain")
       
       theta.all[,i] = chains.current.individual.df[which(chains.current.individual.df$iteration %in% (max(chains.current.individual.df$iteration)-999):max(chains.current.individual.df$iteration)),"toi"]
       
       model.outputs[teller,"toi.estimated.mean"] = round(mean(chains.current.individual.df[,"toi"]))
       model.outputs[teller,"peak.titer.estimated.mean.pomona"] = round(mean(chains.current.individual.df[,"peak.titer.pomona"]),1)
       model.outputs[teller,"decay.rate.estimated.mean.pomona"] = round(mean(chains.current.individual.df[,"decay.rate.pomona"]),6)
       model.outputs[teller,"peak.titer.estimated.mean.aut"] = round(mean(chains.current.individual.df[,"peak.titer.aut"]),1)
       model.outputs[teller,"decay.rate.estimated.mean.aut"] = round(mean(chains.current.individual.df[,"decay.rate.aut"]),6)
       
       model.outputs[teller,"toi.estimated.median"] = median(round(chains.current.individual.df[,"toi"]))
       model.outputs[teller,"peak.titer.estimated.median.pomona"] = median(round(chains.current.individual.df[,"peak.titer.pomona"],1))
       model.outputs[teller,"decay.rate.estimated.median.pomona"] = median(round(chains.current.individual.df[,"decay.rate.pomona"],6))
       model.outputs[teller,"peak.titer.estimated.median.aut"] = median(round(chains.current.individual.df[,"peak.titer.aut"],1))
       model.outputs[teller,"decay.rate.estimated.median.aut"] = median(round(chains.current.individual.df[,"decay.rate.aut"],6))
       
       
       
       dens = density(chains.current.individual.df[,"toi"],bw = 5,from = neg.intervals[i],to=0)
       model.outputs[teller,"toi.estimated.maxdens"] = round(dens$x[which.max(dens$y)])
       hpd.int.toi = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    # highest posterior density credible intervals
       # quantile(chains.current.individual.df[which(chains.current.individual.df$iteration > burn.in),2],c(0.025,0.975))
       model.outputs[teller,"toi.estimated.maxdens.95hdi.low"] = round(hpd.int.toi[1])
       model.outputs[teller,"toi.estimated.maxdens.95hdi.high"] = round(hpd.int.toi[2])
       
       model.outputs[teller,"toi.information.gained.95cri.perc"] = 1-abs((as.numeric(hpd.int.toi[2]-hpd.int.toi[1])/neg.intervals[i]))
       
       hpd.int.toi.80 = HDInterval::hdi(dens,credMass = 0.80,allowSplit=F)    # highest posterior density credible intervals
       model.outputs[teller,"toi.information.gained.80cri.perc"] = 1-abs((as.numeric(hpd.int.toi.80[2]-hpd.int.toi.80[1])/neg.intervals[i]))
       
       hpd.int.toi.50 = HDInterval::hdi(dens,credMass = 0.50,allowSplit=F)    # highest posterior density credible intervals
       model.outputs[teller,"toi.information.gained.50cri.perc"] = 1-abs((as.numeric(hpd.int.toi.50[2]-hpd.int.toi.50[1])/neg.intervals[i]))
       
       ### Relative entropy / Kullback-Leibler divergence
       prior.values = neg.intervals[i]:0
       posterior.values = chains.current.individual.df[,"toi"]
       model.outputs[teller,"kl.divergence"] = kl.divergence(prior.values,posterior.values)
       
       
       
       
       
       mean.eti.toi = mean.eti(chains.current.individual.df[,"toi"])
       model.outputs[teller,"toi.estimated.mean.95ci.low"] = round(mean.eti.toi[1])
       model.outputs[teller,"toi.estimated.mean.95ci.high"] = round(mean.eti.toi[2])
       
       
       
       dens = density(chains.current.individual.df[,"peak.titer.pomona"])
       model.outputs[teller,"peak.titer.estimated.maxdens.pomona"] = round(dens$x[which.max(dens$y)],1)
       hpd.int.peak.titer.pomona = HDInterval::hdi(dens,allowSplit=F)
       
       mean.eti.peak.titer.pomona = mean.eti(chains.current.individual.df[,"peak.titer.pomona"])
       model.outputs[teller,"peak.titer.estimated.maxdens.95hdi.low.pomona"] = round(hpd.int.peak.titer.pomona[1],1)
       model.outputs[teller,"peak.titer.estimated.maxdens.95hdi.high.pomona"] = round(hpd.int.peak.titer.pomona[2],1)
       model.outputs[teller,"peak.titer.estimated.mean.95ci.low.pomona"] = round(mean.eti.peak.titer.pomona[1],1)
       model.outputs[teller,"peak.titer.estimated.mean.95ci.high.pomona"] = round(mean.eti.peak.titer.pomona[2],1)
       
       
       dens = density(chains.current.individual.df[,"peak.titer.aut"])
       model.outputs[teller,"peak.titer.estimated.maxdens.aut"] = round(dens$x[which.max(dens$y)],1)
       hpd.int.peak.titer.aut = HDInterval::hdi(dens,allowSplit=F)
       
       mean.eti.peak.titer.aut = mean.eti(chains.current.individual.df[,"peak.titer.aut"])
       model.outputs[teller,"peak.titer.estimated.maxdens.95hdi.low.aut"] = round(hpd.int.peak.titer.aut[1],1)
       model.outputs[teller,"peak.titer.estimated.maxdens.95hdi.high.aut"] = round(hpd.int.peak.titer.aut[2],1)
       model.outputs[teller,"peak.titer.estimated.mean.95ci.low.aut"] = round(mean.eti.peak.titer.aut[1],1)
       model.outputs[teller,"peak.titer.estimated.mean.95ci.high.aut"] = round(mean.eti.peak.titer.aut[2],1)
       
       
       
       dens = density(chains.current.individual.df[,"decay.rate.pomona"])
       model.outputs[teller,"decay.rate.estimated.maxdens.pomona"] = round(dens$x[which.max(dens$y)],6)
       hpd.int.decay.rate.pomona = HDInterval::hdi(dens,allowSplit=F)
       
       mean.eti.decay.rate.pomona = mean.eti(chains.current.individual.df[,"decay.rate.pomona"])
       model.outputs[teller,"decay.rate.estimated.maxdens.95hdi.low.pomona"] = round(hpd.int.decay.rate.pomona[1],6)
       model.outputs[teller,"decay.rate.estimated.maxdens.95hdi.high.pomona"] = round(hpd.int.decay.rate.pomona[2],6)
       model.outputs[teller,"decay.rate.estimated.mean.95ci.low.pomona"] = round(mean.eti.decay.rate.pomona[1],6)
       model.outputs[teller,"decay.rate.estimated.mean.95ci.high.pomona"] = round(mean.eti.decay.rate.pomona[2],6)
       
       
       dens = density(chains.current.individual.df[,"decay.rate.aut"])
       model.outputs[teller,"decay.rate.estimated.maxdens.aut"] = round(dens$x[which.max(dens$y)],6)
       hpd.int.decay.rate.aut = HDInterval::hdi(dens,allowSplit=F)
       
       mean.eti.decay.rate.aut = mean.eti(chains.current.individual.df[,"decay.rate.aut"])
       model.outputs[teller,"decay.rate.estimated.maxdens.95hdi.low.aut"] = round(hpd.int.decay.rate.aut[1],6)
       model.outputs[teller,"decay.rate.estimated.maxdens.95hdi.high.aut"] = round(hpd.int.decay.rate.aut[2],6)
       model.outputs[teller,"decay.rate.estimated.mean.95ci.low.aut"] = round(mean.eti.decay.rate.aut[1],6)
       model.outputs[teller,"decay.rate.estimated.mean.95ci.high.aut"] = round(mean.eti.decay.rate.aut[2],6)
       
       
       
       
       
       plot1 = ggplot(data=chains.current.individual.df,aes(x=toi)) +
              geom_density(color=brewer.pal(11,"Spectral")[10],fill=brewer.pal(11,"Spectral")[10],alpha=0.6,size=1) +
              geom_vline(xintercept = model.outputs[teller,"toi.estimated.maxdens"],col="black",size=0.8,alpha=0.6) +
              geom_vline(xintercept = model.outputs[teller,"toi.estimated.maxdens.95hdi.low"],col="black",size=0.8,linetype="dotted",alpha=0.6) +
              geom_vline(xintercept = model.outputs[teller,"toi.estimated.maxdens.95hdi.high"],col="black",size=0.8,linetype="dotted",alpha=0.6) +
              theme_light(base_family = "Avenir Next") +
              ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
              xlab("\u03B8 (days)") +
              ylab("Posterior density") +
              theme(plot.title = element_text(size = 11),
                    text=element_text(size=11),
                    axis.text=element_text(size=10),
                    axis.title.y = element_text(margin = ggplot2::margin(r=10)),
                    plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
                    legend.position = "none"
              )
       
       
       
       #plot2 = plot.MCMC.chains(input.data = chains.current.individual.df, column.to.plot = which(colnames(chains.current.individual.df)=="toi"), y.axis.label = "Theta",thinning = 20, title = paste0("Individual ",model.outputs[i,"pittag"]))
       
       if(plot.figs==T) print(plot1)
       
       
       
       
       plot1 = ggplot(data = data.frame(x = 0:15),aes(x=x)) +
              stat_function(fun = dnorm, args = list(mean = model.outputs[teller,"peak.titer.estimated.maxdens.pomona"],sd = sd(chains.current.individual.df$peak.titer.pomona)),aes(col="Pomona"),size=1.5) +
              geom_vline(xintercept = model.outputs[teller,"peak.titer.estimated.maxdens.pomona"],col=brewer.pal(11,"Spectral")[10]) +
              stat_function(fun = dnorm, args = list(mean = model.outputs[teller,"peak.titer.estimated.maxdens.aut"],sd = sd(chains.current.individual.df$peak.titer.aut)),aes(col="Autumnalis"),size=1.5) +
              geom_vline(xintercept = model.outputs[teller,"peak.titer.estimated.maxdens.aut"],col=brewer.pal(11,"Spectral")[1]) +
              scale_color_manual("Serovar",values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
              theme_light(base_family = "Avenir Next") +
              ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
              xlab("Peak antibody level (log2 dilution)") +
              ylab("Posterior density") +
              theme(plot.title = element_text(size = 11),
                    text=element_text(size=11),
                    axis.text=element_text(size=10),
                    axis.title.y = element_text(margin = ggplot2::margin(r=10)),
                    plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
              )
       
       
       
       
       
       
       plot2 = ggplot(data = data.frame(x = seq(-0.00015,0.002,0.0001)),aes(x=x)) +
              stat_function(fun = dnorm, args = list(mean = model.outputs[teller,"decay.rate.estimated.maxdens.pomona"],sd = sd(chains.current.individual.df$decay.rate.pomona)),aes(col="Pomona"),size=1.5) +
              geom_vline(xintercept = model.outputs[teller,"decay.rate.estimated.maxdens.pomona"],col=brewer.pal(11,"Spectral")[10]) +
              stat_function(fun = dnorm, args = list(mean = model.outputs[teller,"decay.rate.estimated.maxdens.aut"],sd = sd(chains.current.individual.df$decay.rate.aut)),aes(col="Autumnalis"),size=1.5) +
              geom_vline(xintercept = model.outputs[teller,"decay.rate.estimated.maxdens.aut"],col=brewer.pal(11,"Spectral")[1]) +
              scale_color_manual("Serovar",values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
              ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
              xlab("Decay rate (1/day)") +
              ylab("Posterior density") +
              theme_light(base_family = "Avenir Next") +
              theme(plot.title = element_text(size = 11),
                    text=element_text(size=11),
                    axis.text=element_text(size=10),
                    axis.title.y = element_text(margin = ggplot2::margin(r=10)),
                    plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
              )
       
       
       
       if(plot.figs==T) print(plot1)
       if(plot.figs==T) print(plot2)
       
       
       
       
       
       
       # pomona
       # sample N values from posterior distribution to get sense of error
       
       N.samp = 200
       
       
       set.seed(1234);peak.titer.samp = sample(chains.current.individual.df$peak.titer.pomona[which(chains.current.individual.df$peak.titer.pomona > hpd.int.peak.titer.pomona[1] & chains.current.individual.df$peak.titer.pomona < hpd.int.peak.titer.pomona[2])], size = N.samp, replace = T)
       
       set.seed(1234);decay.rate.samp = sample(chains.current.individual.df$decay.rate.pomona[which(chains.current.individual.df$decay.rate.pomona > hpd.int.decay.rate.pomona[1] & chains.current.individual.df$decay.rate.pomona < hpd.int.decay.rate.pomona[2])], size = N.samp, replace = T)
       
       plot.times = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)])
       
       fit.dat.cloud.pomona = data.frame(time = rep(plot.times,N.samp), samp = rep(1:N.samp,each = length(plot.times)), titer = NA)
       
       for(ii in 1:N.samp){
              
              fit.dat.cloud.pomona$titer[which(fit.dat.cloud.pomona$samp == ii)] = exp.fun(start.titer = peak.titer.samp[ii], rate = -decay.rate.samp[ii], time = plot.times)
              
       }
       
       
       fit.dat.pomona = exp.fun(start.titer = model.outputs[teller,"peak.titer.estimated.mean.pomona"], rate = -model.outputs[teller,"decay.rate.estimated.mean.pomona"], time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]))
       
       fit.dat.pomona = data.frame(titer = fit.dat.pomona, time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]))
       
       
       plot3 = ggplot() +
              #geom_line(data = observed.data.for.fitting[which(observed.data.for.fitting$id==i),],aes(x = time.since.peak,y = titer),alpha = 0.8,size = 1.5,color="#F46D43") +
              geom_line(data=fit.dat.cloud.pomona, aes(x = time, y = titer, group = samp), col = brewer.pal(11,"Spectral")[10],alpha=0.2,size=0.4) +
              geom_line(data=fit.dat.pomona,aes(x = time, y = titer),col=brewer.pal(11,"Spectral")[10],size=1.5,alpha=1) +
              geom_point(data = observed.data.for.fitting[which(observed.data.for.fitting$id==i),],aes(x = time.since.peak,y = titer.pomona),size=3,alpha = 1,color="black") +
              #scale_color_gradientn(colours  =  brewer.pal(11,"Spectral")) +
              scale_y_continuous(limits = c(0,13),breaks=0:13) +
              ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
              theme_light(base_family = "Avenir Next") +
              theme(plot.title = element_text(size = 11),
                    text=element_text(size=11),
                    axis.text=element_text(size=10),
                    axis.title.y = element_text(margin = ggplot2::margin(r=10)),
                    plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
              ) +
              xlab("Time since peak antibody level (days)") +
              ylab("Antibody level (Pomona) (log2 dilution)")
       
       
       if(plot.figs==T) print(plot3)
       
       
       # aut
       # sample N values from posterior distribution to get sense of error
       
       N.samp = 200
       
       
       set.seed(1234);peak.titer.samp = sample(chains.current.individual.df$peak.titer.aut[which(chains.current.individual.df$peak.titer.aut > hpd.int.peak.titer.aut[1] & chains.current.individual.df$peak.titer.aut < hpd.int.peak.titer.aut[2])], size = N.samp, replace = T)
       
       set.seed(1234);decay.rate.samp = sample(chains.current.individual.df$decay.rate.aut[which(chains.current.individual.df$decay.rate.aut > hpd.int.decay.rate.aut[1] & chains.current.individual.df$decay.rate.aut < hpd.int.decay.rate.aut[2])], size = N.samp, replace = T)
       
       plot.times = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)])
       
       fit.dat.cloud.aut = data.frame(time = rep(plot.times,N.samp), samp = rep(1:N.samp,each = length(plot.times)), titer = NA)
       
       for(ii in 1:N.samp){
              
              fit.dat.cloud.aut$titer[which(fit.dat.cloud.aut$samp == ii)] = exp.fun(start.titer = peak.titer.samp[ii], rate = -decay.rate.samp[ii], time = plot.times)
              
       }
       
       
       if(is.finite(observed.data.for.fitting$titer.aut[which(observed.data.for.fitting$id==i)][1])){
              fit.dat.aut = exp.fun(start.titer = model.outputs[teller,"peak.titer.estimated.mean.aut"], rate = -model.outputs[teller,"decay.rate.estimated.mean.aut"], time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]))
              
              fit.dat.aut = data.frame(titer = fit.dat.aut, time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]))
              
              
              plot4 = ggplot() +
                     #geom_line(data = observed.data.for.fitting[which(observed.data.for.fitting$id==i),],aes(x = time.since.peak,y = titer),alpha = 0.8,size = 1.5,color="#F46D43") +
                     geom_line(data=fit.dat.cloud.aut, aes(x = time, y = titer, group = samp), col = brewer.pal(11,"Spectral")[1],alpha=0.2,size=0.4) +
                     geom_line(data=fit.dat.aut,aes(x = time, y = titer),col=brewer.pal(11,"Spectral")[1],size=1.5,alpha=1) +
                     geom_point(data = observed.data.for.fitting[which(observed.data.for.fitting$id==i),],aes(x = time.since.peak,y = titer.aut),size=3,alpha = 1,color="black") +
                     #scale_color_gradientn(colours  =  brewer.pal(11,"Spectral")) +
                     scale_y_continuous(limits = c(0,13),breaks=0:13) +
                     ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
                     theme_light(base_family = "Avenir Next") +
                     theme(plot.title = element_text(size = 11),
                           text=element_text(size=11),
                           axis.text=element_text(size=10),
                           axis.title.y = element_text(margin = ggplot2::margin(r=10)),
                           plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
                     )+
                     xlab("Time since peak antibody level (days)") +
                     ylab("Antibody level (Autumnalis) (log2 dilution)")
              if(plot.figs==T) print(plot4)
       }
       
       # print(paste0("% seroconversion interval reduction: ",round(model.outputs$toi.information.gained.95cri.perc[teller],4)))
       # print(paste0("KL-divergence: ",round(model.outputs$kl.divergence[teller],4)))
       # print(paste0("Individual ",i," of ",N.inds))
       # print("----------------------------------------------")
       # print("----------------------------------------------")
       # print("----------------------------------------------")
       
       teller = teller + 1
}



3.4 Individual posteriors and chains


Posterior densities are shown after removing burn-in iterations.
Solid red line = estimated posterior maximum density
Weaker red line = estimated posterior mean



Figure for main text:

# model.outputs$pittag[which(model.outputs$toi.information.gained.95cri.perc %in% head(sort(model.outputs$toi.information.gained.95cri.perc)))]
# model.outputs$pittag[which(model.outputs$toi.information.gained.95cri.perc %in% tail(sort(model.outputs$toi.information.gained.95cri.perc)))]

# observed.data.for.fitting[which(observed.data.for.fitting$Pittag=="B6069"),]

pittag.to.plot = c("12672","B6069")



# individual 1
i = observed.data.for.fitting$id[which(observed.data.for.fitting$Pittag==pittag.to.plot[1])][1]
chains.current.individual.df = chains.burn.df[,c(paste0("peak.titer.pomona.",i),paste0("peak.titer.aut.",i),paste0("decay.rate.pomona.",i),paste0("decay.rate.aut.",i),paste0("toi.",i),"iteration","chain")]
colnames(chains.current.individual.df) = c("peak.titer.pomona","peak.titer.aut","decay.rate.pomona","decay.rate.aut","toi","iteration","chain")

cur.peak.titer.pomona = round(mean(chains.current.individual.df[,"peak.titer.pomona"]),1)
dens = density(chains.current.individual.df[,"peak.titer.pomona"])
hpd.int.peak.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    
cur.95.peak.pomona.low = round(hpd.int.peak.pomona[1])
cur.95.peak.pomona.high = round(hpd.int.peak.pomona[2])
hpd.int.peak.pomona = HDInterval::hdi(dens,credMass = 0.5,allowSplit=F)    
cur.50.peak.pomona.low = round(hpd.int.peak.pomona[1])
cur.50.peak.pomona.high = round(hpd.int.peak.pomona[2])


cur.decay.pomona = round(mean(chains.current.individual.df[,"decay.rate.pomona"]),6)
dens = density(chains.current.individual.df[,"decay.rate.pomona"])
hpd.int.decay.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    
cur.95.decay.pomona.low = hpd.int.decay.pomona[1]
cur.95.decay.pomona.high = hpd.int.decay.pomona[2]
hpd.int.decay.pomona = HDInterval::hdi(dens,credMass = 0.50,allowSplit=F)    
cur.50.decay.pomona.low = hpd.int.decay.pomona[1]
cur.50.decay.pomona.high = hpd.int.decay.pomona[2]

cur.peak.titer.aut = round(mean(chains.current.individual.df[,"peak.titer.aut"]),1)
dens = density(chains.current.individual.df[,"peak.titer.aut"])
hpd.int.peak.aut = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    
cur.95.peak.aut.low = round(hpd.int.peak.aut[1])
cur.95.peak.aut.high = round(hpd.int.peak.aut[2])
hpd.int.peak.aut = HDInterval::hdi(dens,credMass = 0.50,allowSplit=F)    
cur.50.peak.aut.low = round(hpd.int.peak.aut[1])
cur.50.peak.aut.high = round(hpd.int.peak.aut[2])


cur.decay.aut = round(mean(chains.current.individual.df[,"decay.rate.aut"]),6)
dens = density(chains.current.individual.df[,"decay.rate.aut"])
hpd.int.decay.aut = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    
cur.95.decay.aut.low = hpd.int.decay.aut[1]
cur.95.decay.aut.high = hpd.int.decay.aut[2]
hpd.int.decay.aut = HDInterval::hdi(dens,credMass = 0.50,allowSplit=F)    
cur.50.decay.aut.low = hpd.int.decay.aut[1]
cur.50.decay.aut.high = hpd.int.decay.aut[2]

dens = density(chains.current.individual.df[,"toi"],bw = 5,from = neg.intervals[i],to=0)
cur.toi = round(dens$x[which.max(dens$y)])
hpd.int.toi = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    
cur.95.low = round(hpd.int.toi[1])
cur.95.high = round(hpd.int.toi[2])
hpd.int.toi = HDInterval::hdi(dens,credMass = 0.50,allowSplit=F)    
cur.50.low = round(hpd.int.toi[1])
cur.50.high = round(hpd.int.toi[2])


plot1a = ggplot(data=chains.current.individual.df,aes(x=toi)) +
       geom_density(color=brewer.pal(4,"Spectral")[2],fill=brewer.pal(4,"Spectral")[2],alpha=0.6,size=1) +
       geom_vline(xintercept = cur.toi,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept = cur.95.low,col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept = cur.95.high,col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept = cur.50.low,col="black",size=0.8,linetype=2,alpha=0.6) +
       geom_vline(xintercept = cur.50.high,col="black",size=0.8,linetype=2,alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       #ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
       xlab("\u03B8 (days)") +
       ylab("Posterior density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             #legend.position = c(0.78,0.85),
             legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       )+
       labs(tag = "(A)")



plot1b = ggplot(data = data.frame(x = 0:15),aes(x=x)) +
       stat_function(fun = dnorm, args = list(mean = cur.peak.titer.pomona,sd = sd(chains.current.individual.df$peak.titer.pomona)),aes(col="Pomona"),size=1.5) +
       #geom_vline(xintercept = cur.peak.titer.pomona,col=brewer.pal(11,"Spectral")[10]) +
       stat_function(fun = dnorm, args = list(mean = cur.peak.titer.aut,sd = sd(chains.current.individual.df$peak.titer.aut)),aes(col="Autumnalis"),size=1.5) +
       #geom_vline(xintercept = cur.peak.titer.aut,col=brewer.pal(11,"Spectral")[1]) +
       scale_color_manual("Serovar",values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
       theme_light(base_family = "Avenir Next") +
       #ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
       xlab("Peak antibody level (log2 dilution)") +
       ylab("Posterior density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = c(0.2,0.8),
             #legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       )

plot1c = ggplot(data = data.frame(x = seq(-0.00015,0.002,0.0001)),aes(x=x)) +
       stat_function(fun = dnorm, args = list(mean = cur.decay.pomona,sd = sd(chains.current.individual.df$decay.rate.pomona)),aes(col="Pomona"),size=1.5) +
       #geom_vline(xintercept = cur.decay.pomona,col=brewer.pal(11,"Spectral")[10]) +
       stat_function(fun = dnorm, args = list(mean = cur.decay.aut,sd = sd(chains.current.individual.df$decay.rate.aut)),aes(col="Autumnalis"),size=1.5) +
       #geom_vline(xintercept = cur.peak.titer.aut,col=brewer.pal(11,"Spectral")[1]) +
       scale_color_manual("Serovar",values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
       theme_light(base_family = "Avenir Next") +
       #ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
       xlab("Decay rate (1/day)") +
       ylab("Posterior density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = c(0.15,0.8),
             #legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       )




# pomona
# sample N values from posterior distribution to get sense of error

N.samp = 200


set.seed(1234);peak.titer.samp.pomona = sample(chains.current.individual.df$peak.titer.pomona[which(chains.current.individual.df$peak.titer.pomona > cur.95.peak.pomona.low & chains.current.individual.df$peak.titer.pomona < cur.95.peak.pomona.high)], size = N.samp, replace = T)

set.seed(1234);decay.rate.samp.pomona = sample(chains.current.individual.df$decay.rate.pomona[which(chains.current.individual.df$decay.rate.pomona > cur.95.decay.pomona.low & chains.current.individual.df$decay.rate.pomona < cur.95.decay.pomona.high)], size = N.samp, replace = T)

set.seed(1234);peak.titer.samp.aut = sample(chains.current.individual.df$peak.titer.aut[which(chains.current.individual.df$peak.titer.aut > cur.95.peak.aut.low & chains.current.individual.df$peak.titer.aut < cur.95.peak.aut.high)], size = N.samp, replace = T)

set.seed(1234);decay.rate.samp.aut = sample(chains.current.individual.df$decay.rate.aut[which(chains.current.individual.df$decay.rate.aut > cur.95.decay.aut.low & chains.current.individual.df$decay.rate.aut < cur.95.decay.aut.high)], size = N.samp, replace = T)


plot.times = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)])

fit.dat.cloud.pomona = data.frame(time = rep(plot.times,N.samp), samp = rep(1:N.samp,each = length(plot.times)), titer = NA,Serovar = "Pomona")

fit.dat.cloud.aut = data.frame(time = rep(plot.times,N.samp), samp = rep(1:N.samp,each = length(plot.times)), titer = NA,Serovar = "Autumnalis")


for(ii in 1:N.samp){
       
       fit.dat.cloud.pomona$titer[which(fit.dat.cloud.pomona$samp == ii)] = exp.fun(start.titer = peak.titer.samp.pomona[ii], rate = -decay.rate.samp.pomona[ii], time = plot.times)
       
       fit.dat.cloud.aut$titer[which(fit.dat.cloud.aut$samp == ii)] = exp.fun(start.titer = peak.titer.samp.aut[ii], rate = -decay.rate.samp.aut[ii], time = plot.times)
       
}

fit.dat.cloud.both = rbind(fit.dat.cloud.pomona,fit.dat.cloud.aut)
fit.dat.cloud.both$samp = paste(fit.dat.cloud.both$Serovar,fit.dat.cloud.both$samp)
fit.dat.cloud.both$Serovar = factor(fit.dat.cloud.both$Serovar,levels = c("Autumnalis","Pomona"))

fit.dat.pomona = exp.fun(start.titer = cur.peak.titer.pomona, rate = -cur.decay.pomona, time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]))

fit.dat.aut = exp.fun(start.titer = cur.peak.titer.aut, rate = -cur.decay.aut, time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]))


fit.dat.pomona = data.frame(titer = fit.dat.pomona, time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]),Serovar = "Pomona")


fit.dat.aut = data.frame(titer = fit.dat.aut, time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]),Serovar = "Autumnalis")

fit.dat.both = rbind(fit.dat.pomona,fit.dat.aut)

cur.observed.data.for.fitting = gather(observed.data.for.fitting,Serovar,titer,c("titer.pomona","titer.aut"))

cur.observed.data.for.fitting$Serovar[which(cur.observed.data.for.fitting$Serovar=="titer.pomona")] = "Pomona"
cur.observed.data.for.fitting$Serovar[which(cur.observed.data.for.fitting$Serovar=="titer.aut")] = "Autumnalis"

# cur.observed.data.for.fitting$Serovar = factor(cur.observed.data.for.fitting$Serovar,levels = c("Pomona","Autumnalis"))

plot1d = ggplot() +
       #geom_line(data = observed.data.for.fitting[which(observed.data.for.fitting$id==i),],aes(x = time.since.peak,y = titer),alpha = 0.8,size = 1.5,color="#F46D43") +
       geom_line(data=fit.dat.cloud.both, aes(x = time, y = titer, group = samp, col = Serovar ),alpha=0.1,size=0.3) +
       geom_line(data=fit.dat.both,aes(x = time, y = titer,col=Serovar),size=1.75,alpha=1) +
       geom_point(data = cur.observed.data.for.fitting[which(cur.observed.data.for.fitting$id==i),],aes(x = time.since.peak,y = titer,shape=Serovar),size=3,alpha = 0.65,color="black") +
       scale_color_manual(name = "Serovar",labels = c("Pomona","Autumnalis"),values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
       scale_shape_manual(name = "Serovar",labels = c("Pomona","Autumnalis"),values = c("Pomona" = 16,"Autumnalis" = 17)) +
       scale_y_continuous(limits = c(0,13),breaks=0:13) +
       #ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = c(0.75,0.82),
             #legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       ) +
       xlab("Time since peak antibody level (days)") +
       ylab("Antibody level (log2 dilution)")





# individual 2
i = observed.data.for.fitting$id[which(observed.data.for.fitting$Pittag==pittag.to.plot[2])][1]
chains.current.individual.df = chains.burn.df[,c(paste0("peak.titer.pomona.",i),paste0("peak.titer.aut.",i),paste0("decay.rate.pomona.",i),paste0("decay.rate.aut.",i),paste0("toi.",i),"iteration","chain")]
colnames(chains.current.individual.df) = c("peak.titer.pomona","peak.titer.aut","decay.rate.pomona","decay.rate.aut","toi","iteration","chain")

cur.peak.titer.pomona = round(mean(chains.current.individual.df[,"peak.titer.pomona"]),1)
dens = density(chains.current.individual.df[,"peak.titer.pomona"])
hpd.int.peak.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    
cur.95.peak.pomona.low = round(hpd.int.peak.pomona[1])
cur.95.peak.pomona.high = round(hpd.int.peak.pomona[2])
hpd.int.peak.pomona = HDInterval::hdi(dens,credMass = 0.5,allowSplit=F)    
cur.50.peak.pomona.low = round(hpd.int.peak.pomona[1])
cur.50.peak.pomona.high = round(hpd.int.peak.pomona[2])


cur.decay.pomona = round(mean(chains.current.individual.df[,"decay.rate.pomona"]),6)
dens = density(chains.current.individual.df[,"decay.rate.pomona"])
hpd.int.decay.pomona = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    
cur.95.decay.pomona.low = hpd.int.decay.pomona[1]
cur.95.decay.pomona.high = hpd.int.decay.pomona[2]
hpd.int.decay.pomona = HDInterval::hdi(dens,credMass = 0.50,allowSplit=F)    
cur.50.decay.pomona.low = hpd.int.decay.pomona[1]
cur.50.decay.pomona.high = hpd.int.decay.pomona[2]

cur.peak.titer.aut = round(mean(chains.current.individual.df[,"peak.titer.aut"]),1)
dens = density(chains.current.individual.df[,"peak.titer.aut"])
hpd.int.peak.aut = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    
cur.95.peak.aut.low = round(hpd.int.peak.aut[1])
cur.95.peak.aut.high = round(hpd.int.peak.aut[2])
hpd.int.peak.aut = HDInterval::hdi(dens,credMass = 0.50,allowSplit=F)    
cur.50.peak.aut.low = round(hpd.int.peak.aut[1])
cur.50.peak.aut.high = round(hpd.int.peak.aut[2])


cur.decay.aut = round(mean(chains.current.individual.df[,"decay.rate.aut"]),6)
dens = density(chains.current.individual.df[,"decay.rate.aut"])
hpd.int.decay.aut = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    
cur.95.decay.aut.low = hpd.int.decay.aut[1]
cur.95.decay.aut.high = hpd.int.decay.aut[2]
hpd.int.decay.aut = HDInterval::hdi(dens,credMass = 0.50,allowSplit=F)    
cur.50.decay.aut.low = hpd.int.decay.aut[1]
cur.50.decay.aut.high = hpd.int.decay.aut[2]

dens = density(chains.current.individual.df[,"toi"],bw = 5,from = neg.intervals[i],to=0)
cur.toi = round(dens$x[which.max(dens$y)])
hpd.int.toi = HDInterval::hdi(dens,credMass = 0.95,allowSplit=F)    
cur.95.low = round(hpd.int.toi[1])
cur.95.high = round(hpd.int.toi[2])
hpd.int.toi = HDInterval::hdi(dens,credMass = 0.50,allowSplit=F)    
cur.50.low = round(hpd.int.toi[1])
cur.50.high = round(hpd.int.toi[2])


plot2a = ggplot(data=chains.current.individual.df,aes(x=toi)) +
       geom_density(color=brewer.pal(4,"Spectral")[2],fill=brewer.pal(4,"Spectral")[2],alpha=0.6,size=1) +
       geom_vline(xintercept = cur.toi,col="black",size=0.8,alpha=0.6) +
       geom_vline(xintercept = cur.95.low,col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept = cur.95.high,col="black",size=0.8,linetype="dotted",alpha=0.6) +
       geom_vline(xintercept = cur.50.low,col="black",size=0.8,linetype=2,alpha=0.6) +
       geom_vline(xintercept = cur.50.high,col="black",size=0.8,linetype=2,alpha=0.6) +
       theme_light(base_family = "Avenir Next") +
       #ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
       xlab("\u03B8 (days)") +
       ylab("Posterior density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             #legend.position = c(0.78,0.85),
             legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       ) +
       labs(tag = "(B)")



plot2b = ggplot(data = data.frame(x = 0:15),aes(x=x)) +
       stat_function(fun = dnorm, args = list(mean = cur.peak.titer.pomona,sd = sd(chains.current.individual.df$peak.titer.pomona)),aes(col="Pomona"),size=1.5) +
       #geom_vline(xintercept = cur.peak.titer.pomona,col=brewer.pal(11,"Spectral")[10]) +
       stat_function(fun = dnorm, args = list(mean = cur.peak.titer.aut,sd = sd(chains.current.individual.df$peak.titer.aut)),aes(col="Autumnalis"),size=1.5) +
       #geom_vline(xintercept = cur.peak.titer.aut,col=brewer.pal(11,"Spectral")[1]) +
       scale_color_manual("Serovar",values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
       theme_light(base_family = "Avenir Next") +
       #ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
       xlab("Peak antibody level (log2 dilution)") +
       ylab("Posterior density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = c(0.2,0.8),
             #legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       )

plot2c = ggplot(data = data.frame(x = seq(-0.00015,0.002,0.0001)),aes(x=x)) +
       stat_function(fun = dnorm, args = list(mean = cur.decay.pomona,sd = sd(chains.current.individual.df$decay.rate.pomona)),aes(col="Pomona"),size=1.5) +
       #geom_vline(xintercept = cur.decay.pomona,col=brewer.pal(11,"Spectral")[10]) +
       stat_function(fun = dnorm, args = list(mean = cur.decay.aut,sd = sd(chains.current.individual.df$decay.rate.aut)),aes(col="Autumnalis"),size=1.5) +
       #geom_vline(xintercept = cur.peak.titer.aut,col=brewer.pal(11,"Spectral")[1]) +
       scale_color_manual("Serovar",values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
       theme_light(base_family = "Avenir Next") +
       #ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
       xlab("Decay rate (1/day)") +
       ylab("Posterior density") +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = c(0.15,0.8),
             #legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       )




# pomona
# sample N values from posterior distribution to get sense of error

N.samp = 200


set.seed(1234);peak.titer.samp.pomona = sample(chains.current.individual.df$peak.titer.pomona[which(chains.current.individual.df$peak.titer.pomona > cur.95.peak.pomona.low & chains.current.individual.df$peak.titer.pomona < cur.95.peak.pomona.high)], size = N.samp, replace = T)

set.seed(1234);decay.rate.samp.pomona = sample(chains.current.individual.df$decay.rate.pomona[which(chains.current.individual.df$decay.rate.pomona > cur.95.decay.pomona.low & chains.current.individual.df$decay.rate.pomona < cur.95.decay.pomona.high)], size = N.samp, replace = T)

set.seed(1234);peak.titer.samp.aut = sample(chains.current.individual.df$peak.titer.aut[which(chains.current.individual.df$peak.titer.aut > cur.95.peak.aut.low & chains.current.individual.df$peak.titer.aut < cur.95.peak.aut.high)], size = N.samp, replace = T)

set.seed(1234);decay.rate.samp.aut = sample(chains.current.individual.df$decay.rate.aut[which(chains.current.individual.df$decay.rate.aut > cur.95.decay.aut.low & chains.current.individual.df$decay.rate.aut < cur.95.decay.aut.high)], size = N.samp, replace = T)


plot.times = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)])

fit.dat.cloud.pomona = data.frame(time = rep(plot.times,N.samp), samp = rep(1:N.samp,each = length(plot.times)), titer = NA,Serovar = "Pomona")

fit.dat.cloud.aut = data.frame(time = rep(plot.times,N.samp), samp = rep(1:N.samp,each = length(plot.times)), titer = NA,Serovar = "Autumnalis")


for(ii in 1:N.samp){
       
       fit.dat.cloud.pomona$titer[which(fit.dat.cloud.pomona$samp == ii)] = exp.fun(start.titer = peak.titer.samp.pomona[ii], rate = -decay.rate.samp.pomona[ii], time = plot.times)
       
       fit.dat.cloud.aut$titer[which(fit.dat.cloud.aut$samp == ii)] = exp.fun(start.titer = peak.titer.samp.aut[ii], rate = -decay.rate.samp.aut[ii], time = plot.times)
       
}

fit.dat.cloud.both = rbind(fit.dat.cloud.pomona,fit.dat.cloud.aut)
fit.dat.cloud.both$samp = paste(fit.dat.cloud.both$Serovar,fit.dat.cloud.both$samp)
fit.dat.cloud.both$Serovar = factor(fit.dat.cloud.both$Serovar,levels = c("Autumnalis","Pomona"))

fit.dat.pomona = exp.fun(start.titer = cur.peak.titer.pomona, rate = -cur.decay.pomona, time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]))

fit.dat.aut = exp.fun(start.titer = cur.peak.titer.aut, rate = -cur.decay.aut, time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]))


fit.dat.pomona = data.frame(titer = fit.dat.pomona, time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]),Serovar = "Pomona")


fit.dat.aut = data.frame(titer = fit.dat.aut, time = 0:max(observed.data.for.fitting$time.since.peak[which(observed.data.for.fitting$id==i)]),Serovar = "Autumnalis")

fit.dat.both = rbind(fit.dat.pomona,fit.dat.aut)

cur.observed.data.for.fitting = gather(observed.data.for.fitting,Serovar,titer,c("titer.pomona","titer.aut"))

cur.observed.data.for.fitting$Serovar[which(cur.observed.data.for.fitting$Serovar=="titer.pomona")] = "Pomona"
cur.observed.data.for.fitting$Serovar[which(cur.observed.data.for.fitting$Serovar=="titer.aut")] = "Autumnalis"

# cur.observed.data.for.fitting$Serovar = factor(cur.observed.data.for.fitting$Serovar,levels = c("Pomona","Autumnalis"))

plot2d = ggplot() +
       #geom_line(data = observed.data.for.fitting[which(observed.data.for.fitting$id==i),],aes(x = time.since.peak,y = titer),alpha = 0.8,size = 1.5,color="#F46D43") +
       geom_line(data=fit.dat.cloud.both, aes(x = time, y = titer, group = samp, col = Serovar ),alpha=0.1,size=0.3) +
       geom_line(data=fit.dat.both,aes(x = time, y = titer,col=Serovar),size=1.75,alpha=1) +
       geom_point(data = cur.observed.data.for.fitting[which(cur.observed.data.for.fitting$id==i),],aes(x = time.since.peak,y = titer,shape=Serovar),size=3,alpha = 0.65,color="black") +
       scale_color_manual(name = "Serovar",labels = c("Pomona","Autumnalis"),values = c("Pomona" = brewer.pal(11,"Spectral")[10],"Autumnalis" = brewer.pal(11,"Spectral")[1])) +
       scale_shape_manual(name = "Serovar",labels = c("Pomona","Autumnalis"),values = c("Pomona" = 16,"Autumnalis" = 17)) +
       scale_y_continuous(limits = c(0,13),breaks=0:13) +
       #ggtitle(paste0("Individual ",model.outputs[i,"pittag"])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 13),
             text=element_text(size=13),
             axis.text=element_text(size=13),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             legend.position = c(0.3,0.2),
             #legend.position = "none",
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.background = element_rect(colour = "transparent", fill = "transparent"),
             legend.key = element_rect(fill = NA)
       ) +
       xlab("Time since peak antibody level (days)") +
       ylab("Antibody level (log2 dilution)")



plot_fig4 = plot1a + plot1b + plot1d + 
       plot2a + plot2b + plot2d

plot_fig4

3.5 Individual posteriors for seroconversion time

Using full posterior density, 80% CrI and 50% CrI:

#plot.order = rev(order(model.outputs$toi.estimated.maxdens))  # order by maxdens
plot.order = rev(order(neg.intervals))  # order by maxdens

theta.all.ordered = as.data.frame(theta.all[,plot.order]) %>%
       gather

theta.all.ordered$key = rep(1:N.inds, each = 6000)
colnames(theta.all.ordered) = c("id","toi")

theta.all.ordered$toi50 = theta.all.ordered$toi
theta.all.ordered$toi80 = theta.all.ordered$toi
theta.all.ordered$toi95 = theta.all.ordered$toi

for(i in 1:N.inds){
       toi.sorted = sort(theta.all.ordered$toi[which(theta.all.ordered$id==unique(theta.all.ordered$id)[i])])
       
       toi.50ci.low = toi.sorted[floor(0.25 * length(toi.sorted))]
       toi.50ci.hi = toi.sorted[floor(0.75 * length(toi.sorted))]
       theta.all.ordered$toi50[which(theta.all.ordered$id==unique(theta.all.ordered$id)[i] & (theta.all.ordered$toi < toi.50ci.low | theta.all.ordered$toi > toi.50ci.hi))] = NA
       
       toi.80ci.low = toi.sorted[floor(0.1 * length(toi.sorted))]
       toi.80ci.hi = toi.sorted[floor(0.9 * length(toi.sorted))]
       theta.all.ordered$toi80[which(theta.all.ordered$id==unique(theta.all.ordered$id)[i] & (theta.all.ordered$toi < toi.80ci.low | theta.all.ordered$toi > toi.80ci.hi))] = NA
       
       toi.95ci.low = toi.sorted[floor(0.025 * length(toi.sorted))]
       toi.95ci.hi = toi.sorted[floor(0.975 * length(toi.sorted))]
       theta.all.ordered$toi95[which(theta.all.ordered$id==unique(theta.all.ordered$id)[i] & (theta.all.ordered$toi < toi.95ci.low | theta.all.ordered$toi > toi.95ci.hi))] = NA
}

# divide into 4 parts for plotting
theta.all.ordered.part1 = theta.all.ordered[which(theta.all.ordered$id %in% 1:floor(N.inds/4)),]
theta.all.ordered.part2 = theta.all.ordered[which(theta.all.ordered$id %in% (floor(N.inds/4)+1):(floor(N.inds/4)*2)),]
theta.all.ordered.part3 = theta.all.ordered[which(theta.all.ordered$id %in% ((floor(N.inds/4)*2)+1):(floor(N.inds/4)*3)),]
theta.all.ordered.part4 = theta.all.ordered[which(theta.all.ordered$id %in% ((floor(N.inds/4)*3)+1):N.inds),]


individual.theta.plots.1 = ggplot() +
       geom_density_ridges(data = theta.all.ordered.part1,aes(x = toi,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[10],alpha=0.8) +
       geom_density_ridges(data = theta.all.ordered.part1,aes(x = toi80,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[2],alpha=0.8) +
       geom_density_ridges(data = theta.all.ordered.part1,aes(x = toi50,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[5],alpha=0.8) +
       coord_flip() +
       ylab("ID") +
       xlab("\u03B8 (days)") +
       #ggtitle("1/4") +
       theme_light(base_family = "Avenir Next") +        
       theme(plot.title  =  element_text(size  = 11),              
             text = element_text(size = 11),              
             axis.text.y = element_text(size = 11),
             axis.text.x = element_blank(),        
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)))

individual.theta.plots.2 = ggplot() +
       geom_density_ridges(data = theta.all.ordered.part2,aes(x = toi,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[10],alpha=0.8) +
       geom_density_ridges(data = theta.all.ordered.part2,aes(x = toi80,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[2],alpha=0.8) +
       geom_density_ridges(data = theta.all.ordered.part2,aes(x = toi50,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[5],alpha=0.8) +
       coord_flip() +
       ylab("ID") +
       xlab("\u03B8 (days)") +
       #ggtitle("Peak titer time (2/4)") +
       theme_light(base_family = "Avenir Next") +        
       theme(plot.title  =  element_text(size  = 11),              
             text = element_text(size = 11),              
             axis.text.y = element_text(size = 11),
             axis.text.x = element_blank(),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10))              )

individual.theta.plots.3 = ggplot() +
       geom_density_ridges(data = theta.all.ordered.part3,aes(x = toi,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[10],alpha=0.8) +
       geom_density_ridges(data = theta.all.ordered.part3,aes(x = toi80,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[2],alpha=0.8) +
       geom_density_ridges(data = theta.all.ordered.part3,aes(x = toi50,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[5],alpha=0.8) +
       coord_flip() +
       ylab("ID") +
       xlab("\u03B8 (days)") +
       #ggtitle("Peak titer time (3/4)") +
       theme_light(base_family = "Avenir Next") +        
       theme(plot.title  =  element_text(size  = 11),              
             text = element_text(size = 11),              
             axis.text.y = element_text(size = 11),
             axis.text.x = element_blank(),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10))              )

individual.theta.plots.4 = ggplot() +
       geom_density_ridges(data = theta.all.ordered.part4,aes(x = toi,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[10],alpha=0.8) +
       geom_density_ridges(data = theta.all.ordered.part4,aes(x = toi80,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[2],alpha=0.8) +
       geom_density_ridges(data = theta.all.ordered.part4,aes(x = toi50,y = factor(id),height=..density..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[5],alpha=0.8) +
       coord_flip() +
       ylab("ID") +
       xlab("\u03B8 (days)") +
       #ggtitle("Peak titer time (4/4)") +
       theme_light(base_family = "Avenir Next") +        
       theme(plot.title  =  element_text(size  = 11),              
             text = element_text(size = 11),              
             axis.text.y = element_text(size = 11),
             axis.text.x = element_blank(),          
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10))              )

individual.theta.plots.all = individual.theta.plots.1 / 
       individual.theta.plots.2 / 
       individual.theta.plots.3 /
       individual.theta.plots.4

individual.theta.plots.all

Using full posterior density, 80% CrI and 50% CrI:

individual.theta.plots.1 = ggplot() +
       geom_density_ridges(data = theta.all.ordered.part1,aes(x = toi,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[10],alpha=0.8, scale = 0.8) +
       geom_density_ridges(data = theta.all.ordered.part1,aes(x = toi95,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[2],alpha=0.8, scale = 0.8) +
       geom_density_ridges(data = theta.all.ordered.part1,aes(x = toi50,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[5],alpha=0.8, scale = 0.8) +
       coord_flip() +
       ylab("ID") +
       xlab("\u03B8 (days)") +
       #ggtitle("1/4") +
       theme_light(base_family = "Avenir Next") +        
       theme(plot.title  =  element_text(size  = 11),              
             text = element_text(size = 11),              
             axis.text.y = element_text(size = 11),
             axis.text.x = element_blank(),        
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10)))

individual.theta.plots.2 = ggplot() +
       geom_density_ridges(data = theta.all.ordered.part2,aes(x = toi,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[10],alpha=0.8, scale = 0.8) +
       geom_density_ridges(data = theta.all.ordered.part2,aes(x = toi95,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[2],alpha=0.8, scale = 0.8) +
       geom_density_ridges(data = theta.all.ordered.part2,aes(x = toi50,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[5],alpha=0.8, scale = 0.8) +
       coord_flip() +
       ylab("ID") +
       xlab("\u03B8 (days)") +
       #ggtitle("Peak titer time (2/4)") +
       theme_light(base_family = "Avenir Next") +        
       theme(plot.title  =  element_text(size  = 11),              
             text = element_text(size = 11),              
             axis.text.y = element_text(size = 11),
             axis.text.x = element_blank(),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10))              )

individual.theta.plots.3 = ggplot() +
       geom_density_ridges(data = theta.all.ordered.part3,aes(x = toi,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[10],alpha=0.8, scale = 0.8) +
       geom_density_ridges(data = theta.all.ordered.part3,aes(x = toi95,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[2],alpha=0.8, scale = 0.8) +
       geom_density_ridges(data = theta.all.ordered.part3,aes(x = toi50,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[5],alpha=0.8, scale = 0.8) +
       coord_flip() +
       ylab("ID") +
       xlab("\u03B8 (days)") +
       #ggtitle("Peak titer time (3/4)") +
       theme_light(base_family = "Avenir Next") +        
       theme(plot.title  =  element_text(size  = 11),              
             text = element_text(size = 11),              
             axis.text.y = element_text(size = 11),
             axis.text.x = element_blank(),
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10))              )

individual.theta.plots.4 = ggplot() +
       geom_density_ridges(data = theta.all.ordered.part4,aes(x = toi,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[10],alpha=0.8, scale = 0.8) +
       geom_density_ridges(data = theta.all.ordered.part4,aes(x = toi95,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[2],alpha=0.8, scale = 0.8) +
       geom_density_ridges(data = theta.all.ordered.part4,aes(x = toi50,y = factor(id),height=..scaled..), stat = "density",size=0.1,trim=T,fill = brewer.pal(11,"Spectral")[5],alpha=0.8, scale = 0.8) +
       coord_flip() +
       ylab("ID") +
       xlab("\u03B8 (days)") +
       #ggtitle("Peak titer time (4/4)") +
       theme_light(base_family = "Avenir Next") +        
       theme(plot.title  =  element_text(size  = 11),              
             text = element_text(size = 11),              
             axis.text.y = element_text(size = 11),
             axis.text.x = element_blank(),          
             axis.title.y  =  element_text(margin  =  ggplot2::margin(r = 10))              )

individual.theta.plots.all = individual.theta.plots.1 / 
       individual.theta.plots.2 / 
       individual.theta.plots.3 /
       individual.theta.plots.4

individual.theta.plots.all

# ggsave(plot = individual.theta.plots.all,filename = "/Users/bennyborremans/Documents/Werk/Manuscripten/Eerste auteur/ongoing/Fox titer kinetics/Figures/Figure_posterior_toi_all_individuals_incl95_50cri.png",width = 9, height = 12, dpi=600)




3.6 Mean information gained

3.6.1 Using only individuals for which there is at least one Autumnalis

result

# exclude individuals without any Aut samples     

model.outputs.aut.not.all.na = model.outputs[id.aut.not.all.na,]

Mean amount of information gained (% reduction of seroconversion interval):

95% CrI:

0.2391879
(standard error = 0.0150179)
Range: 0.0547945, 0.81409

Median:
0.130137

80% CrI: 0.4087182
(standard error = 0.0123609)
Range: 0.2093933, 0.8454012 Median:
0.3522505

50% CrI: 0.6627593
(standard error = 0.0076397)
Range: 0.5088063, 0.9354207 Median:
0.646771
## KL-divergence

3.6.2 Using only individuals for which there is at least one Autumnalis

result

Mean amount of information gained (KL-divergence):
0.262891
(standard error = 0.0488155)
Range: -0.5260461, 2.3497259

Median: 0.262891

Histogram:

plot.seroconv.red = ggplot(data = model.outputs.aut.not.all.na, aes(x = toi.information.gained.95cri.perc)) +
       geom_histogram(bins=12,binwidth = 0.01,fill="#9E0142",col="black",aes(y = ..density..)) +
       theme_classic(base_family = "Avenir Next") +
       scale_x_continuous(limits = c(0,0.6)) +
       scale_color_manual("Distribution",values = c("N(6,2.2)" = "#66C2A5","N(7,3)" = "#FDAE61")) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Seroconversion interval reduction (%)") +
       ylab("Density")

plot.seroconv.red

# ggsave(plot = plot.seroconv.red,filename = "/Users/bennyborremans/Documents/Werk/Manuscripten/Eerste auteur/ongoing/Fox titer kinetics/Figures/SI_figure_seroconversion_interval_reduction_at_least_one_aut_pos_histogram.png",width = 4, height = 4, dpi=600)

3.7 Mean information gained

3.7.1 Using all individuals

Mean amount of information gained (% reduction of seroconversion interval):

95% CrI: 0.2317293
(standard error = 0.0120415)
Range: 0.0547945, 0.81409 Median:
0.1252446

80% CrI: 0.4016969
(standard error = 0.0099033)
Range: 0.2093933, 0.8454012 Median:
0.3483366

50% CrI: 0.6581143
(standard error = 0.0061543)
Range: 0.5088063, 0.9354207 Median:
0.6418787

3.8 KL-divergence

3.8.1 Using all individuals

Mean amount of information gained (KL-divergence):
0.2473702
(standard error = 0.0389285)
Range: -0.5260461, 2.3497259

Median:
0.2473702

Histogram:

95% CrI:

plot.seroconv.red = ggplot(data = model.outputs, aes(x = toi.information.gained.95cri.perc)) +
       geom_histogram(bins=12,binwidth = 0.01,fill="#9E0142",col="black",aes(y = ..density..)) +
       theme_classic(base_family = "Avenir Next") +
       scale_x_continuous(breaks = seq(0,1,0.1)) +
       scale_color_manual("Distribution",values = c("N(6,2.2)" = "#66C2A5","N(7,3)" = "#FDAE61")) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Seroconversion interval reduction (%)") +
       ylab("Density")

plot.seroconv.red

# ggsave(plot = plot.seroconv.red,filename = "/Users/bennyborremans/Documents/Werk/Manuscripten/Eerste auteur/ongoing/Fox titer kinetics/Figures/SI_figure_seroconversion_interval_reduction_95cri_all_samples_histogram.png",width = 4, height = 4, dpi=600)

80% CrI:

plot.seroconv.red = ggplot(data = model.outputs, aes(x = toi.information.gained.80cri.perc)) +
       geom_histogram(bins=12,binwidth = 0.01,fill="#9E0142",col="black",aes(y = ..density..)) +
       theme_classic(base_family = "Avenir Next") +
       scale_x_continuous(breaks = seq(0,1,0.1)) +
       scale_color_manual("Distribution",values = c("N(6,2.2)" = "#66C2A5","N(7,3)" = "#FDAE61")) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Seroconversion interval reduction (%)") +
       ylab("Density")

plot.seroconv.red

# ggsave(plot = plot.seroconv.red,filename = "/Users/bennyborremans/Documents/Werk/Manuscripten/Eerste auteur/ongoing/Fox titer kinetics/Figures/SI_figure_seroconversion_interval_reduction_80cri_all_samples_histogram.png",width = 4, height = 4, dpi=600)

50% CrI:

plot.seroconv.red = ggplot(data = model.outputs, aes(x = toi.information.gained.50cri.perc)) +
       geom_histogram(bins=12,binwidth = 0.01,fill="#9E0142",col="black",aes(y = ..density..)) +
       theme_classic(base_family = "Avenir Next") +
       scale_x_continuous(breaks = seq(0,1,0.1)) +
       scale_color_manual("Distribution",values = c("N(6,2.2)" = "#66C2A5","N(7,3)" = "#FDAE61")) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Seroconversion interval reduction (%)") +
       ylab("Density")

plot.seroconv.red

# ggsave(plot = plot.seroconv.red,filename = "/Users/bennyborremans/Documents/Werk/Manuscripten/Eerste auteur/ongoing/Fox titer kinetics/Figures/SI_figure_seroconversion_interval_reduction_50cri_all_samples_histogram.png",width = 4, height = 4, dpi=600)

3.9 Correlation between interval size reduction and KL-divergence

95% CrI:

ggplot(model.outputs,aes(x = toi.information.gained.95cri.perc, y = kl.divergence)) +
       geom_point(color = brewer.pal(11,"Spectral")[2],size = 0.5) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Seroconversion interval reduction (%)") +
       ylab("Relative entropy (bits)")

80% CrI:

ggplot(model.outputs,aes(x = toi.information.gained.80cri.perc, y = kl.divergence)) +
       geom_point(color = brewer.pal(11,"Spectral")[2],size = 0.5) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Seroconversion interval reduction (%)") +
       ylab("Relative entropy (bits)")

50% CrI:

ggplot(model.outputs,aes(x = toi.information.gained.50cri.perc, y = kl.divergence)) +
       geom_point(color = brewer.pal(11,"Spectral")[2],size = 0.5) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Seroconversion interval reduction (%)") +
       ylab("Relative entropy (bits)")

3.10 Correlates of model performance

Why does the model result in better estimates of seroconversion time for some individuals than for others?

3.10.1 Seroconversion interval size

All for 95% CrI

model.outputs$interval.size = abs(neg.intervals)

ggplot(model.outputs,aes(x = interval.size, y = 100*toi.information.gained.95cri.perc)) +
       geom_point(color = brewer.pal(11,"Spectral")[2],size = 0.7) +
       scale_y_continuous(limits=c(0,65)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       )+
       xlab("Seroconversion interval size (days)") +
       ylab("Interval reduction (%)")

# lm.interval.size = lm(log(toi.information.gained.95cri.perc) ~ scale(interval.size),data = model.outputs)

lm.interval.size = brm(log(toi.information.gained.95cri.perc) ~ scale(interval.size),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
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## Chain 2: 
## 
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## Chain 3: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 4).
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## Chain 4:

3.10.2 Range of time covered by datapoints

model.outputs$datarange = NA

for(i in 1:nrow(model.outputs)){
       model.outputs$datarange[i] = max(observed.data.for.fitting$time[which(observed.data.for.fitting$id == i)])
}

ggplot(model.outputs,aes(x = datarange, y = 100*toi.information.gained.95cri.perc)) +
       geom_point(color = brewer.pal(11,"Spectral")[2],size = 0.5) +
       scale_y_continuous(limits=c(0,65)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Range of datapoints (days)") +
       ylab("Interval reduction (%)")

# lm.datarange = lm(log(toi.information.gained.95cri.perc) ~ scale(datarange),data = model.outputs)
lm.datarange = brm(log(toi.information.gained.95cri.perc) ~ scale(datarange),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 1.9e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.19 seconds.
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## Chain 1: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
## Chain 2: 
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## Chain 2: Adjust your expectations accordingly!
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## Chain 2:                0.039975 seconds (Total)
## Chain 2: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 6e-06 seconds
## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
## Chain 3: Adjust your expectations accordingly!
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## Chain 3: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 4).
## Chain 4: 
## Chain 4: Gradient evaluation took 1.2e-05 seconds
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## Chain 4:

3.10.3 Number of datapoints

model.outputs$number.of.datapoints = NA

for(i in 1:nrow(model.outputs)){
       model.outputs$number.of.datapoints[i] = sum(observed.data.for.fitting$id == i)
}

ggplot(model.outputs,aes(x = number.of.datapoints, y = 100*toi.information.gained.95cri.perc)) +
       geom_point(color = brewer.pal(11,"Spectral")[2],size = 0.5) +
       scale_y_continuous(limits = c(0,65)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Number of datapoints") +
       ylab("Interval reduction (%)")

#lm.number.of.datapoints = lm(log(toi.information.gained.95cri.perc) ~ scale(number.of.datapoints),data = model.outputs)
lm.number.of.datapoints = brm(log(toi.information.gained.95cri.perc) ~ scale(number.of.datapoints),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
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## Chain 2: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 4).
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## Chain 4:

3.10.4 Estimated peak level

ggplot(model.outputs,aes(x = peak.titer.estimated.mean.pomona, y = 100*toi.information.gained.95cri.perc)) +
       geom_point(color = brewer.pal(11,"Spectral")[2],size = 0.5) +
       scale_y_continuous(limits = c(0,65)) +
       scale_x_continuous(breaks = seq(0,15,2)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Estimated peak antibody level \n(log2 dilution)") +
       ylab("Interval reduction (%)")

#lm.estimated.peak.level = lm(log(toi.information.gained.95cri.perc) ~ scale(peak.titer.estimated.mean.pomona),data = model.outputs)
lm.estimated.peak.level = brm(log(toi.information.gained.95cri.perc) ~ scale(peak.titer.estimated.mean.pomona),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 5.4e-05 seconds
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## Chain 1: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
## Chain 2: 
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## Chain 2: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 6e-06 seconds
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## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 4).
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## Chain 4: Gradient evaluation took 5e-06 seconds
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## Chain 4:

3.10.5 Antibody level of the first positive sample

model.outputs$first.pos.level = observed.data.for.fitting$titer.pomona[which(observed.data.for.fitting$time==0)]

# unique(observed.data.for.fitting$Pittag) %in% model.outputs$pittag

ggplot(model.outputs,aes(x = first.pos.level, y = 100*toi.information.gained.95cri.perc)) +
       geom_jitter(color = brewer.pal(11,"Spectral")[2],size = 0.7,width=0.1) +
       scale_x_continuous(breaks=seq(0,15,2)) +
       scale_y_continuous(limits=c(0,65)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       )+
       xlab("Pomona level first positive sample \n(log2 dilutions)") +
       ylab("Interval reduction (%)")

#lm.first.pos.level = lm(log(toi.information.gained.95cri.perc) ~ scale(first.pos.level),data = model.outputs)
lm.first.pos.level = brm(log(toi.information.gained.95cri.perc) ~ scale(first.pos.level),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 0.000388 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 3.88 seconds.
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## Chain 1: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
## Chain 2: 
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## Chain 2: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
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## Chain 3: Adjust your expectations accordingly!
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 4).
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## Chain 4:

3.10.6 Estimated decay rate

ggplot(model.outputs,aes(x = decay.rate.estimated.mean.pomona, y = 100*toi.information.gained.95cri.perc)) +
       geom_point(color = brewer.pal(11,"Spectral")[2],size = 0.5) +
       scale_y_continuous(limits = c(0,65)) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Estimated decay rate") +
       ylab("Interval reduction (%)")

#lm.decay.rate = lm(log(toi.information.gained.95cri.perc) ~ scale(decay.rate.estimated.mean.pomona),data = model.outputs)
lm.decay.rate = brm(log(toi.information.gained.95cri.perc) ~ scale(decay.rate.estimated.mean.pomona),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
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## Chain 2: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
## Chain 3: 
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 4).
## Chain 4: 
## Chain 4: Gradient evaluation took 8e-06 seconds
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## Chain 4:

3.10.7 Estimated decay rate and estimate peak antibody level

ggplot(model.outputs,aes(x = decay.rate.estimated.mean.pomona, y = peak.titer.estimated.mean.pomona,color = 100*toi.information.gained.95cri.perc)) +
       geom_point(size = 1) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       scale_color_gradientn(name = "Interval \nreduction \n(%)",colors =  rev(brewer.pal(11,"Spectral")[c(1,2,3,5,8,10,11)])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Estimated decay rate") +
       ylab("Estimated peak level")

#lm.decay.rate.and.estimated.peak.level = lm(log(toi.information.gained.95cri.perc) ~ scale(decay.rate.estimated.mean.pomona) + scale(peak.titer.estimated.mean.pomona),data = model.outputs)
lm.decay.rate.and.estimated.peak.level = brm(log(toi.information.gained.95cri.perc) ~ scale(decay.rate.estimated.mean.pomona) + scale(peak.titer.estimated.mean.pomona),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 2e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.2 seconds.
## Chain 1: Adjust your expectations accordingly!
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## Chain 1:                0.051117 seconds (Total)
## Chain 1: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
## Chain 2: 
## Chain 2: Gradient evaluation took 7e-06 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.07 seconds.
## Chain 2: Adjust your expectations accordingly!
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## Chain 2: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 6e-06 seconds
## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
## Chain 3: Adjust your expectations accordingly!
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 4).
## Chain 4: 
## Chain 4: Gradient evaluation took 7e-06 seconds
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## Chain 4:

3.10.8 Estimated decay rate and first positive level

ggplot(model.outputs,aes(x = decay.rate.estimated.mean.pomona, y = first.pos.level,color = 100*toi.information.gained.95cri.perc)) +
       geom_point(size = 1) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       scale_color_gradientn(name = "Interval \nreduction \n(%)",colors =  rev(brewer.pal(11,"Spectral")[c(1,2,3,5,8,10,11)])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Estimated decay rate") +
       ylab("First positive sample level")

#lm.decay.rate.and.first.positive.level = lm(log(toi.information.gained.95cri.perc) ~ scale(decay.rate.estimated.mean.pomona) + scale(first.pos.level),data = model.outputs)
lm.decay.rate.and.first.positive.level = brm(log(toi.information.gained.95cri.perc) ~ scale(decay.rate.estimated.mean.pomona) + scale(first.pos.level),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 0.00028 seconds
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## Chain 1: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
## Chain 2: 
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 4).
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## Chain 4:

3.10.9 Interval size and decay rate

ggplot(model.outputs,aes(x = decay.rate.estimated.mean.pomona, y = interval.size,color = 100*toi.information.gained.95cri.perc)) +
       geom_point(size = 1) +
       #scale_y_continuous(breaks = seq(0,15,2)) +
       scale_color_gradientn(name = "Interval \nreduction \n(%)",colors =  rev(brewer.pal(11,"Spectral")[c(1,2,3,5,8,10,11)])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Estimated decay rate (1/day)") +
       ylab("Seroconversion interval size (days)")

lm.decay.rate.and.interval.size = brm(log(toi.information.gained.95cri.perc) ~ scale(decay.rate.estimated.mean.pomona) + scale(interval.size),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 2e-05 seconds
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## Chain 1: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
## Chain 2: 
## Chain 2: Gradient evaluation took 8e-06 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.08 seconds.
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## Chain 2: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 8e-06 seconds
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 4).
## Chain 4: 
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## Chain 4:

3.10.10 Interval size and peak level

ggplot(model.outputs,aes(x = interval.size, y = peak.titer.estimated.mean.pomona,color = 100*toi.information.gained.95cri.perc)) +
       geom_point(size = 1) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       scale_color_gradientn(name = "Interval \nreduction \n(%)",colors =  rev(brewer.pal(11,"Spectral")[c(1,2,3,5,8,10,11)])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Seroconversion interval size (days)") +
       ylab("Estimated peak level (log2 dilution)")

lm.peaklevel.and.interval.size = brm(log(toi.information.gained.95cri.perc) ~ scale(peak.titer.estimated.mean.pomona) + scale(interval.size),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 6.6e-05 seconds
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## Chain 1: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
## Chain 2: 
## Chain 2: Gradient evaluation took 7e-06 seconds
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## Chain 2: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
## Chain 3: 
## Chain 3: Gradient evaluation took 7e-06 seconds
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 4).
## Chain 4: 
## Chain 4: Gradient evaluation took 6e-06 seconds
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## Chain 4:

3.10.11 Estimated decay, estimated peak, interval size

plot.interval.decay = ggplot(model.outputs,aes(x = interval.size, y = decay.rate.estimated.mean.pomona,color = 100*toi.information.gained.95cri.perc)) +
       geom_point(size = 1) +
       #scale_y_continuous(breaks = seq(0,15,2)) +
       scale_color_gradientn(name = "Interval \nreduction \n(%)",colors =  rev(brewer.pal(11,"Spectral")[c(1,2,3,5,8,10,11)])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = 'none'
       ) +
       ylab("Estimated decay rate (1/day)") +
       xlab("Infection window (days)")

plot.interval.peak = ggplot(model.outputs,aes(x = interval.size, y = peak.titer.estimated.mean.pomona,color = 100*toi.information.gained.95cri.perc)) +
       geom_point(size = 1) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       scale_color_gradientn(name = "Interval \nreduction \n(%)",colors =  rev(brewer.pal(11,"Spectral")[c(1,2,3,5,8,10,11)])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = 'none'
       ) +
       xlab("Infection window (days)") +
       ylab("Estimated peak level \n(log2 dilution)")

plot.interval.firstpos = ggplot(model.outputs,aes(x = interval.size, y = first.pos.level,color = 100*toi.information.gained.95cri.perc)) +
       geom_point(size = 1) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       scale_color_gradientn(name = "Interval \nreduction \n(%)",colors =  rev(brewer.pal(11,"Spectral")[c(1,2,3,5,8,10,11)])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm"),
             legend.position = 'none'
       ) +
       xlab("Infection window (days)") +
       ylab("First positive Pomona sample \nlevel (log2 dilution)")



plot.firstpos.decay = ggplot(model.outputs,aes(x = decay.rate.estimated.mean.pomona, y = first.pos.level,color = 100*toi.information.gained.95cri.perc)) +
       geom_point(size = 1) +
       scale_y_continuous(breaks = seq(0,15,2)) +
       scale_color_gradientn(name = "Interval \nreduction \n(%)",colors =  rev(brewer.pal(11,"Spectral")[c(1,2,3,5,8,10,11)])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Estimated decay rate (1/day)") +
       ylab("First positive Pomona sample \nlevel (log2 dilution)")


plot.covariates.reduction = plot.interval.decay + plot.interval.firstpos + plot.firstpos.decay

plot.covariates.reduction

3.10.12 Number of data points and first positive level

ggplot(model.outputs,aes(x = number.of.datapoints, y = first.pos.level,color = 100*toi.information.gained.95cri.perc)) +
       geom_jitter(size = 1,width=0.1) +
       scale_x_continuous(breaks = 0:10) +
       scale_y_continuous(breaks = 0:15) +
       scale_color_gradientn(name = "Interval \nreduction \n(%)",colors =  rev(brewer.pal(11,"Spectral")[c(1,2,3,5,8,10,11)])) +
       theme_light(base_family = "Avenir Next") +
       theme(plot.title = element_text(size = 11),
             text=element_text(size=11),
             axis.text=element_text(size=10),
             axis.title.y = element_text(margin = ggplot2::margin(r=10)),
             plot.margin = unit(c(0.5,0.75,0.5,0.5),"cm")
       ) +
       xlab("Number of positive datapoints") +
       ylab("First positive sample level \n(log2 dilution)")

#lm.number.of.datapoints.and.first.positive.level = lm(log(toi.information.gained.95cri.perc) ~ scale(number.of.datapoints) + scale(first.pos.level),data = model.outputs)
lm.number.of.datapoints.and.first.positive.level = brm(log(toi.information.gained.95cri.perc) ~ scale(number.of.datapoints) + scale(first.pos.level),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
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4 All covariates

lm.all = brm(log(toi.information.gained.95cri.perc) ~ scale(first.pos.level) + scale(decay.rate.estimated.mean.pomona) + scale(peak.titer.estimated.mean.pomona) + scale(interval.size) + scale(datarange) + scale(number.of.datapoints), data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
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4.0.1 Model comparison

4.0.1.1 Univariate models P values

# frequentist:    

# lm.interval.size
# lm.datarange
# lm.number.of.datapoints
# lm.estimated.peak.level
# lm.first.pos.level
# lm.decay.rate
# lm.decay.rate.and.estimated.peak.level
# lm.decay.rate.and.first.positive.level
# lm.number.of.datapoints.and.first.positive.level


# model.stats = data.frame(
#        variables = c("interval size","datarange","number of datapoints","estimated peak","first pos level","decay rate"),
#        Pval = c(anova(lm.interval.size)$Pr[1],anova(lm.datarange)$Pr[1],anova(lm.number.of.datapoints)$Pr[1],anova(lm.estimated.peak.level)$Pr[1],anova(lm.first.pos.level)$Pr[1],anova(lm.decay.rate)$Pr[1]),
#        Fval = round(c(anova(lm.interval.size)$F[1],anova(lm.datarange)$F[1],anova(lm.number.of.datapoints)$F[1],anova(lm.estimated.peak.level)$F[1],anova(lm.first.pos.level)$F[1],anova(lm.decay.rate)$F[1]),2),
#        df = c(anova(lm.interval.size)$Df[2],anova(lm.datarange)$Df[2],anova(lm.number.of.datapoints)$Df[2],anova(lm.estimated.peak.level)$Df[2],anova(lm.first.pos.level)$Df[2],anova(lm.decay.rate)$Df[2]),
#        Effect.estimate.exp = exp(round(c(lm.interval.size$coefficients[2],lm.datarange$coefficients[2],lm.number.of.datapoints$coefficients[2],lm.estimated.peak.level$coefficients[2],lm.first.pos.level$coefficients[2],lm.decay.rate$coefficients[2]),4)),
#        AIC = round(AIC(lm.interval.size,lm.datarange,lm.number.of.datapoints,lm.estimated.peak.level,lm.first.pos.level,lm.decay.rate),1))
# 
# model.stats$Pval = round(model.stats$Pval,6)



# bayesian models:

model.stats = data.frame(
       variables = c("interval size","datarange","number of datapoints","estimated peak","first pos level","decay rate"),
       LOOIC.est = round(c(loo(lm.interval.size)$estimates[3,1],
                           loo(lm.datarange)$estimates[3,1],
                           loo(lm.number.of.datapoints)$estimates[3,1],
                           loo(lm.estimated.peak.level)$estimates[3,1],
                           loo(lm.first.pos.level)$estimates[3,1],
                           loo(lm.decay.rate)$estimates[3,1]),1),
       LOOIC.se = round(c(loo(lm.interval.size)$estimates[3,2],
                          loo(lm.datarange)$estimates[3,2],
                          loo(lm.number.of.datapoints)$estimates[3,2],
                          loo(lm.estimated.peak.level)$estimates[3,2],
                          loo(lm.first.pos.level)$estimates[3,2],
                          loo(lm.decay.rate)$estimates[3,2]),1),
       Effect.estimate.exp = round(c(exp(summary(lm.interval.size)$fixed[2,1]),
                                     exp(summary(lm.datarange)$fixed[2,1]),
                                     exp(summary(lm.number.of.datapoints)$fixed[2,1]),
                                     exp(summary(lm.estimated.peak.level)$fixed[2,1]),
                                     exp(summary(lm.first.pos.level)$fixed[2,1]),
                                     exp(summary(lm.decay.rate)$fixed[2,1])),2),
       Effect.estimate.95lo.exp = round(c(exp(summary(lm.interval.size)$fixed[2,3]),
                                          exp(summary(lm.datarange)$fixed[2,3]),
                                          exp(summary(lm.number.of.datapoints)$fixed[2,3]),
                                          exp(summary(lm.estimated.peak.level)$fixed[2,3]),
                                          exp(summary(lm.first.pos.level)$fixed[2,3]),
                                          exp(summary(lm.decay.rate)$fixed[2,3])),2),
       Effect.estimate.95hi.exp = round(c(exp(summary(lm.interval.size)$fixed[2,4]),
                                          exp(summary(lm.datarange)$fixed[2,4]),
                                          exp(summary(lm.number.of.datapoints)$fixed[2,4]),
                                          exp(summary(lm.estimated.peak.level)$fixed[2,4]),
                                          exp(summary(lm.first.pos.level)$fixed[2,4]),
                                          exp(summary(lm.decay.rate)$fixed[2,4])),2))


kbl(model.stats) %>%
       kable_classic(bootstrap_options  =  c("striped", "hover","condensed"), full_width=F,html_font="Cambria",fixed_thead=T)
variables LOOIC.est LOOIC.se Effect.estimate.exp Effect.estimate.95lo.exp Effect.estimate.95hi.exp
interval size 629.5 22.6 1.65 1.53 1.78
datarange 766.5 16.2 1.01 0.92 1.11
number of datapoints 766.2 16.3 1.03 0.94 1.13
estimated peak 761.5 16.9 1.11 1.01 1.22
first pos level 754.5 17.0 1.17 1.07 1.28
decay rate 752.0 16.6 1.19 1.09 1.31

4.0.1.2 LOOIC values

All for 95% CrI

lm.interval.size.and.decay = brm(log(toi.information.gained.95cri.perc) ~ scale(interval.size) + scale(decay.rate.estimated.mean.pomona),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
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## Chain 1: 
## 
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## Chain 2: 
## 
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## Chain 4:
lm.interval.size.and.peak = brm(log(toi.information.gained.95cri.perc) ~ scale(interval.size) + scale(peak.titer.estimated.mean.pomona),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
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## Chain 1: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
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## Chain 2: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
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## Chain 4:
lm.interval.size.and.peak.and.decay = brm(log(toi.information.gained.95cri.perc) ~ scale(interval.size) + scale(peak.titer.estimated.mean.pomona) + scale(decay.rate.estimated.mean.pomona),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
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## Chain 1: 
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
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## 
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## Chain 4:
lm.number.of.datapoints.and.peak.and.decay = brm(log(toi.information.gained.95cri.perc) ~ scale(number.of.datapoints) + scale(peak.titer.estimated.mean.pomona) + scale(decay.rate.estimated.mean.pomona),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
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## 
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 3).
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lm.number.of.datapoints.and.peak = brm(log(toi.information.gained.95cri.perc) ~ scale(number.of.datapoints) + scale(peak.titer.estimated.mean.pomona),data = model.outputs)
## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## clang -mmacosx-version-min=10.13 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG   -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/Rcpp/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/unsupported"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/src/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppParallel/include/"  -I"/Library/Frameworks/R.framework/Versions/4.2/Resources/library/rstan/include" -DEIGEN_NO_DEBUG  -DBOOST_DISABLE_ASSERTS  -DBOOST_PENDING_INTEGER_LOG2_HPP  -DSTAN_THREADS  -DBOOST_NO_AUTO_PTR  -include '/Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp'  -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1   -I/usr/local/include   -fPIC  -Wall -g -O2  -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
##                ^
##                ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:13:
## In file included from /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.2/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
##          ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1
## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 1).
## Chain 1: 
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## 
## SAMPLING FOR MODEL '530d7032dfbce0764fc0a1dc5bd35928' NOW (CHAIN 2).
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## 
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model.stats = data.frame(
       variables = c("interval size",
                     "datarange",
                     "number of datapoints",
                     "estimated peak",
                     "first pos level",
                     "decay rate",
                     "decay rate + estimated peak",
                     "decay rate + first pos",
                     "N datapoints + peak",
                     "N datapoints + decay + peak",
                     "interval size + decay rate",
                     "interval size + estimated peak",
                     "interval size + decay + peak",
                     "all variables"),
       LOOIC.est = round(c(loo(lm.interval.size)$estimates[3,1],
                           loo(lm.datarange)$estimates[3,1],
                           loo(lm.number.of.datapoints)$estimates[3,1],
                           loo(lm.estimated.peak.level)$estimates[3,1],
                           loo(lm.first.pos.level)$estimates[3,1],
                           loo(lm.decay.rate)$estimates[3,1],
                           loo(lm.decay.rate.and.estimated.peak.level)$estimates[3,1],
                           loo(lm.decay.rate.and.first.positive.level)$estimates[3,1],
                           loo(lm.number.of.datapoints.and.peak)$estimates[3,1],
                           loo(lm.number.of.datapoints.and.peak.and.decay)$estimates[3,1],
                           loo(lm.interval.size.and.decay)$estimates[3,1],
                           loo(lm.interval.size.and.peak)$estimates[3,1],
                           loo(lm.interval.size.and.peak.and.decay)$estimates[3,1],
                           loo(lm.all)$estimates[3,1]),1),
       LOOIC.se = round(c(loo(lm.interval.size)$estimates[3,2],
                          loo(lm.datarange)$estimates[3,2],
                          loo(lm.number.of.datapoints)$estimates[3,2],
                          loo(lm.estimated.peak.level)$estimates[3,2],
                          loo(lm.first.pos.level)$estimates[3,2],
                          loo(lm.decay.rate)$estimates[3,2],
                          loo(lm.decay.rate.and.estimated.peak.level)$estimates[3,2],
                          loo(lm.decay.rate.and.first.positive.level)$estimates[3,2],
                          loo(lm.number.of.datapoints.and.peak)$estimates[3,2],
                          loo(lm.number.of.datapoints.and.peak.and.decay)$estimates[3,2],
                          loo(lm.interval.size.and.decay)$estimates[3,2],
                          loo(lm.interval.size.and.peak)$estimates[3,2],
                          loo(lm.interval.size.and.peak.and.decay)$estimates[3,2],
                          loo(lm.all)$estimates[3,2]),1))

model.stats.ordered = model.stats %>%
       arrange(LOOIC.est)

kbl(model.stats.ordered) %>%
       kable_classic(bootstrap_options  =  c("striped", "hover","condensed"), full_width=F,html_font="Cambria",fixed_thead=T)
variables LOOIC.est LOOIC.se
all variables 576.4 31.3
interval size + decay + peak 596.0 28.9
interval size + decay rate 611.5 24.1
interval size + estimated peak 621.9 24.2
interval size 629.5 22.6
decay rate + first pos 737.9 18.8
decay rate + estimated peak 744.7 18.6
N datapoints + decay + peak 745.8 18.7
decay rate 752.0 16.6
first pos level 754.5 17.0
estimated peak 761.5 16.9
N datapoints + peak 763.0 17.1
number of datapoints 766.2 16.3
datarange 766.5 16.2

4.1 Model output table


kbl(model.outputs) %>%
       kable_classic(bootstrap_options  =  c("striped", "hover","condensed"), full_width=F,html_font="Cambria",fixed_thead=T)
id pittag toi.mid.interval toi.estimated.mean toi.estimated.median toi.estimated.maxdens toi.estimated.mean.95ci.low toi.estimated.mean.95ci.high toi.estimated.maxdens.95hdi.low toi.estimated.maxdens.95hdi.high toi.information.gained.95cri.perc toi.information.gained.80cri.perc toi.information.gained.50cri.perc kl.divergence peak.titer.estimated.mean.pomona peak.titer.estimated.median.pomona peak.titer.estimated.maxdens.pomona peak.titer.estimated.mean.95ci.low.pomona peak.titer.estimated.mean.95ci.high.pomona peak.titer.estimated.maxdens.95hdi.low.pomona peak.titer.estimated.maxdens.95hdi.high.pomona decay.rate.estimated.mean.pomona decay.rate.estimated.median.pomona decay.rate.estimated.maxdens.pomona decay.rate.estimated.mean.95ci.low.pomona decay.rate.estimated.mean.95ci.high.pomona decay.rate.estimated.maxdens.95hdi.low.pomona decay.rate.estimated.maxdens.95hdi.high.pomona peak.titer.estimated.mean.aut peak.titer.estimated.median.aut peak.titer.estimated.maxdens.aut peak.titer.estimated.mean.95ci.low.aut peak.titer.estimated.mean.95ci.high.aut peak.titer.estimated.maxdens.95hdi.low.aut peak.titer.estimated.maxdens.95hdi.high.aut decay.rate.estimated.mean.aut decay.rate.estimated.median.aut decay.rate.estimated.maxdens.aut decay.rate.estimated.mean.95ci.low.aut decay.rate.estimated.mean.95ci.high.aut decay.rate.estimated.maxdens.95hdi.low.aut decay.rate.estimated.maxdens.95hdi.high.aut interval.size datarange number.of.datapoints first.pos.level
1 00141 -136 -160 -172.0 -258 -268 -178 -272 -25 0.0919765 0.3072407 0.6281800 0.0773043 4.3 4.3 4.3 2.4 4.2 2.4 6.3 0.000989 0.0009830 0.000971 0.000368 0.000963 0.000357 0.001637 4.5 4.5 4.5 2.1 4.4 2.1 6.9 0.000660 0.0006570 0.000661 0.000112 0.000640 0.000110 0.001230 272 586 2 3
2 00701 -362 -335 -326.0 -271 -693 -340 -665 -4 0.0880626 0.2778865 0.5733855 0.0363930 7.0 6.9 6.7 5.0 6.8 4.8 9.4 0.000879 0.0008680 0.000842 0.000489 0.000855 0.000470 0.001310 7.5 7.3 7.1 5.2 7.3 5.0 10.3 0.000756 0.0007490 0.000741 0.000309 0.000735 0.000299 0.001229 725 1093 4 5
3 00978 -115 -132 -140.0 -217 -226 -145 -230 -20 0.0861057 0.2896282 0.6066536 0.0551859 4.8 4.8 4.7 3.2 4.7 3.1 6.7 0.001033 0.0010270 0.001046 0.000432 0.001008 0.000414 0.001653 5.1 5.0 5.0 2.9 5.0 2.8 7.4 0.000660 0.0006580 0.000657 0.000111 0.000641 0.000103 0.001216 230 629 3 4
4 00D05 -126 -123 -122.0 -86 -245 -128 -242 -6 0.0665362 0.2250489 0.5225049 0.0086119 8.9 8.9 8.9 7.4 8.9 7.4 10.5 0.000617 0.0006100 0.000598 0.000414 0.000603 0.000402 0.000843 9.9 9.9 9.8 8.3 9.8 8.2 11.7 0.000399 0.0003960 0.000387 0.000237 0.000391 0.000232 0.000569 252 2038 5 7
5 01334 -247 -239 -235.0 -153 -478 -246 -467 -6 0.0665362 0.2387476 0.5401174 0.0121951 7.3 7.2 7.0 5.2 7.1 5.1 9.6 0.000774 0.0007700 0.000786 0.000211 0.000752 0.000202 0.001353 7.8 7.7 7.6 5.3 7.6 5.2 10.5 0.000660 0.0006550 0.000635 0.000109 0.000639 0.000109 0.001224 494 380 2 6
6 01520 -243 -282 -298.0 -444 -477 -309 -486 -45 0.0919765 0.3091977 0.6144814 0.0644158 5.9 5.8 5.6 3.8 5.7 3.7 8.3 0.000862 0.0008590 0.000868 0.000253 0.000840 0.000253 0.001490 6.3 6.2 6.0 3.7 6.1 3.6 9.1 0.000660 0.0006570 0.000696 0.000111 0.000639 0.000102 0.001219 486 360 2 4
7 01619 -148 -169 -178.0 -282 -290 -185 -295 -25 0.0841487 0.2876712 0.6046967 0.0513243 5.2 5.1 5.0 3.4 5.1 3.3 7.2 0.000896 0.0008920 0.000866 0.000318 0.000873 0.000306 0.001489 5.5 5.4 5.4 3.3 5.4 3.2 7.9 0.000659 0.0006570 0.000640 0.000107 0.000640 0.000098 0.001218 295 699 3 4
8 01744 -180 -203 -214.0 -348 -354 -222 -361 -30 0.0821918 0.2778865 0.5929550 0.0406363 5.6 5.6 5.5 4.1 5.5 4.0 7.3 0.000926 0.0009160 0.000872 0.000434 0.000900 0.000415 0.001458 6.0 5.9 5.9 4.3 5.9 4.2 7.9 0.000627 0.0006210 0.000609 0.000214 0.000608 0.000202 0.001063 361 1052 3 4
9 02107 -178 -122 -105.0 -17 -320 -111 -294 0 0.1722114 0.4324853 0.7025440 0.2337434 9.4 9.3 9.1 7.8 9.2 7.7 11.2 0.001118 0.0011110 0.001095 0.000784 0.001100 0.000769 0.001481 10.3 10.2 10.0 8.5 10.1 8.4 12.3 0.000677 0.0006730 0.000655 0.000406 0.000664 0.000397 0.000960 355 1092 4 7
10 03466 -243 -242 -241.0 -237 -472 -253 -466 -10 0.0626223 0.2211350 0.5185910 0.0069562 6.6 6.6 6.4 5.1 6.5 5.0 8.4 0.000579 0.0005740 0.000581 0.000142 0.000561 0.000130 0.001032 7.1 7.0 6.9 5.4 7.0 5.3 9.1 0.000522 0.0005230 0.000545 0.000038 0.000508 0.000041 0.001003 486 694 3 4
11 03662 -194 -264 -276.0 -389 -465 -286 -389 -31 0.0802348 0.2720157 0.5812133 -0.2584606 5.7 5.7 5.6 4.0 5.6 3.9 7.7 0.000886 0.0008720 0.000870 0.000462 0.000857 0.000432 0.001359 6.0 6.0 5.8 4.0 5.9 4.0 8.2 0.000706 0.0007020 0.000693 0.000173 0.000685 0.000166 0.001256 389 2116 4 4
12 0433A -237 -350 -374.0 -474 -535 -384 -474 -66 0.1389432 0.4070450 0.6927593 -0.1303779 5.6 5.6 5.6 4.0 5.5 3.9 7.4 0.001025 0.0010140 0.000999 0.000514 0.000997 0.000491 0.001571 6.0 6.0 5.7 4.3 5.9 4.2 8.0 0.000445 0.0004390 0.000430 0.000039 0.000426 0.000027 0.000870 474 1043 3 3
13 04471 -271 -124 -138.0 -160 -168 -140 -176 -32 0.7338552 0.8434442 0.9354207 2.1991727 7.3 7.3 7.4 5.5 7.3 5.5 9.1 0.001264 0.0012570 0.001237 0.000521 0.001232 0.000504 0.002030 8.5 8.5 8.5 6.6 8.5 6.6 10.5 0.000394 0.0004040 0.000401 -0.000231 0.000385 -0.000209 0.000987 542 0 1 2
14 04703 -84 -121 -112.0 -14 -270 -119 -161 -3 0.0626223 0.2367906 0.5381605 -0.3386416 8.3 8.2 8.1 6.7 8.1 6.6 10.0 0.001094 0.0010840 0.001067 0.000801 0.001074 0.000782 0.001421 8.9 8.9 8.7 7.2 8.8 7.1 10.8 0.000746 0.0007410 0.000725 0.000430 0.000731 0.000422 0.001080 169 2533 8 7
15 04953 -141 -121 -125.0 -209 -218 -130 -221 -13 0.2602740 0.4050881 0.6516634 0.3610675 5.8 5.8 5.7 4.5 5.8 4.5 7.2 0.001086 0.0010760 0.001063 0.000655 0.001062 0.000641 0.001556 6.2 6.2 6.1 4.8 6.1 4.8 7.8 0.000532 0.0005280 0.000539 0.000184 0.000517 0.000177 0.000898 282 1359 5 5
16 05735 -112 -69 -68.0 -14 -140 -71 -138 -3 0.3933464 0.5009785 0.6986301 0.6420731 8.4 8.4 8.4 7.1 8.4 7.1 9.8 0.000726 0.0007170 0.000689 0.000455 0.000708 0.000437 0.001030 9.2 9.2 9.2 7.7 9.1 7.7 10.8 0.000557 0.0005520 0.000528 0.000178 0.000540 0.000168 0.000951 223 2285 4 7
17 05824 -72 -183 -160.0 -32 -463 -170 -141 -6 0.0626223 0.2191781 0.5225049 -0.5110433 8.5 8.4 8.2 6.7 8.3 6.6 10.5 0.000934 0.0009190 0.000890 0.000619 0.000908 0.000597 0.001301 9.2 9.1 9.0 7.1 9.0 7.0 11.5 0.000636 0.0006340 0.000652 0.000106 0.000617 0.000100 0.001170 144 1825 4 7
18 06263 -253 -247 -256.0 -380 -445 -266 -454 -31 0.1643836 0.3346380 0.6046967 0.1780865 6.5 6.4 6.2 4.4 6.3 4.3 8.8 0.000867 0.0008620 0.000864 0.000285 0.000845 0.000276 0.001464 6.9 6.8 6.7 4.4 6.7 4.3 9.7 0.000660 0.0006580 0.000644 0.000107 0.000640 0.000107 0.001223 506 406 2 5
19 06436 -228 -251 -269.0 -409 -416 -279 -424 -37 0.1487280 0.3581213 0.6594912 0.1946071 4.6 4.5 4.4 3.1 4.5 3.0 6.2 0.000713 0.0007060 0.000695 0.000247 0.000691 0.000228 0.001205 4.6 4.5 4.4 2.8 4.5 2.7 6.6 0.000874 0.0008660 0.000872 0.000388 0.000850 0.000379 0.001390 455 1060 4 3
20 06441 -211 -284 -302.0 -402 -473 -312 -422 -42 0.0998043 0.3228963 0.6301370 -0.1610078 6.1 6.0 5.8 3.8 5.9 3.7 8.7 0.001094 0.0010890 0.001044 0.000489 0.001069 0.000481 0.001721 6.5 6.4 6.2 3.7 6.3 3.6 9.5 0.000660 0.0006580 0.000681 0.000111 0.000641 0.000105 0.001225 422 351 2 5
21 06978 -240 -143 -123.0 -15 -363 -130 -341 0 0.2915851 0.4970646 0.7436399 0.4676216 8.4 8.3 8.2 6.8 8.2 6.7 10.1 0.000831 0.0008230 0.000799 0.000436 0.000810 0.000419 0.001252 8.9 8.9 8.8 7.2 8.8 7.1 11.0 0.000640 0.0006370 0.000603 0.000206 0.000623 0.000197 0.001083 481 736 3 8
22 07127 -195 -442 -464.0 -390 -716 -479 -390 -45 0.1154599 0.3502935 0.6497065 -0.4875424 6.0 5.9 5.8 3.7 5.8 3.6 8.5 0.001108 0.0010940 0.001057 0.000672 0.001079 0.000647 0.001591 6.3 6.3 6.0 3.8 6.2 3.7 9.2 0.000693 0.0006900 0.000673 0.000185 0.000674 0.000181 0.001216 390 2184 5 3
23 07478 -364 -242 -218.0 -107 -579 -230 -544 0 0.2544031 0.4559687 0.7025440 0.3532983 7.8 7.7 7.5 6.0 7.6 5.8 10.0 0.000766 0.0007560 0.000726 0.000310 0.000741 0.000295 0.001260 8.4 8.3 8.0 6.3 8.2 6.2 10.9 0.000699 0.0006910 0.000674 0.000290 0.000678 0.000279 0.001133 729 578 2 6
24 07516 -311 -277 -285.0 -380 -498 -297 -508 -35 0.2387476 0.3953033 0.6399217 0.3115895 6.5 6.4 6.2 4.4 6.4 4.3 9.0 0.000874 0.0008700 0.000854 0.000288 0.000851 0.000282 0.001482 7.0 6.9 6.7 4.4 6.8 4.3 9.9 0.000659 0.0006560 0.000665 0.000106 0.000639 0.000104 0.001227 622 365 2 5
25 07620 -255 -60 -58.0 -13 -125 -61 -123 -2 0.7632094 0.8082192 0.8864971 1.9969878 8.2 8.2 8.1 6.5 8.2 6.5 10.1 0.001157 0.0011510 0.001130 0.000578 0.001132 0.000566 0.001755 8.9 8.8 8.8 6.7 8.8 6.7 11.1 0.000658 0.0006570 0.000662 0.000104 0.000639 0.000097 0.001220 510 351 2 9
26 07906 -64 -72 -70.0 -13 -148 -73 -125 -4 0.0606654 0.2289628 0.5342466 -0.1617922 8.4 8.4 8.2 6.8 8.3 6.7 10.1 0.000813 0.0008090 0.000769 0.000386 0.000794 0.000376 0.001258 9.0 9.0 8.9 7.0 8.9 6.9 11.2 0.000659 0.0006550 0.000637 0.000108 0.000638 0.000095 0.001215 129 733 3 8
27 11709 -76 -75 -74.0 -48 -150 -78 -147 -4 0.0684932 0.2270059 0.5264188 0.0036062 7.8 7.8 7.7 6.2 7.7 6.1 9.4 0.000738 0.0007340 0.000730 0.000302 0.000720 0.000293 0.001185 8.4 8.3 8.3 6.3 8.3 6.3 10.5 0.000660 0.0006570 0.000645 0.000107 0.000640 0.000105 0.001226 153 735 3 7
28 11726 -77 -186 -189.0 -137 -350 -198 -154 -10 0.0665362 0.2250489 0.5225049 -0.5233889 6.2 6.2 6.1 4.8 6.1 4.8 7.7 0.000618 0.0006070 0.000585 0.000369 0.000598 0.000351 0.000903 6.5 6.5 6.4 4.9 6.4 4.8 8.3 0.000621 0.0006190 0.000595 0.000132 0.000604 0.000125 0.001114 154 2559 6 4
29 12672 -179 -194 -194.0 -171 -375 -204 -353 -15 0.0547945 0.2093933 0.5088063 -0.0910527 6.0 6.0 5.9 4.7 5.9 4.7 7.4 0.000445 0.0004310 0.000411 0.000056 0.000418 0.000034 0.000888 6.3 6.3 6.2 4.9 6.2 4.8 7.9 0.000374 0.0003670 0.000354 -0.000011 0.000354 -0.000026 0.000788 358 1172 3 5
30 13062 -192 -364 -376.0 -380 -644 -390 -385 -29 0.0763209 0.2798434 0.5909980 -0.4710185 6.1 6.0 5.7 4.3 6.0 4.2 8.3 0.000802 0.0007970 0.000792 0.000300 0.000781 0.000289 0.001327 6.5 6.4 6.2 4.5 6.3 4.4 8.9 0.000629 0.0006280 0.000630 0.000135 0.000613 0.000131 0.001129 385 536 3 3
31 13737 -330 -189 -162.0 -50 -509 -172 -459 0 0.3033268 0.5322896 0.7514677 0.4808743 8.1 8.0 7.8 6.3 7.9 6.2 10.3 0.001036 0.0010240 0.000997 0.000530 0.001008 0.000512 0.001584 8.8 8.6 8.4 6.7 8.6 6.6 11.3 0.000781 0.0007720 0.000742 0.000350 0.000758 0.000338 0.001246 659 435 2 7
32 14353 -310 -74 -73.0 -18 -149 -77 -147 -4 0.7690802 0.8082192 0.8845401 2.0263671 7.6 7.6 7.5 5.9 7.6 5.9 9.5 0.001079 0.0010730 0.001075 0.000496 0.001054 0.000482 0.001684 8.2 8.2 8.2 6.1 8.1 6.0 10.5 0.000659 0.0006560 0.000637 0.000110 0.000639 0.000105 0.001222 621 356 2 8
33 14732 -290 -70 -70.0 -115 -135 -74 -135 -5 0.7749511 0.8140900 0.8864971 2.0754471 7.9 7.8 7.8 6.5 7.8 6.5 9.3 0.001116 0.0011050 0.001086 0.000677 0.001090 0.000658 0.001592 8.6 8.6 8.4 7.1 8.5 7.0 10.1 0.000559 0.0005520 0.000542 0.000225 0.000541 0.000215 0.000920 580 1113 3 7
34 14A0D -76 -192 -172.0 -13 -453 -183 -147 -3 0.0606654 0.2250489 0.5362035 -0.5100509 9.3 9.3 9.3 8.0 9.2 7.9 10.7 0.000338 0.0003370 0.000327 0.000193 0.000332 0.000190 0.000489 10.1 10.1 10.0 8.7 10.0 8.6 11.8 0.000281 0.0002800 0.000281 0.000112 0.000274 0.000110 0.000458 153 1949 7 8
35 15627 -69 -288 -305.0 -138 -484 -316 -138 -11 0.0763209 0.2524462 0.5577299 -0.4325183 6.0 5.9 5.5 3.8 5.8 3.7 8.5 0.000936 0.0009310 0.000924 0.000388 0.000914 0.000378 0.001503 6.4 6.3 6.1 3.7 6.2 3.6 9.3 0.000660 0.0006560 0.000658 0.000107 0.000639 0.000103 0.001224 138 756 3 4
36 15660 -238 -116 -97.0 -13 -302 -104 -284 0 0.4031311 0.5831703 0.7925636 0.7394113 9.6 9.6 9.5 8.0 9.5 7.9 11.4 0.000740 0.0007350 0.000702 0.000071 0.000714 0.000065 0.001420 10.4 10.3 10.2 8.3 10.3 8.2 12.7 0.000659 0.0006570 0.000679 0.000106 0.000640 0.000100 0.001216 475 90 2 9
37 15690 -246 -112 -91.0 -14 -322 -97 -291 0 0.4090020 0.6164384 0.8121331 0.7768594 9.5 9.5 9.2 8.0 9.4 7.9 11.3 0.000855 0.0008490 0.000841 0.000579 0.000840 0.000566 0.001157 10.3 10.2 10.1 8.5 10.2 8.4 12.3 0.000662 0.0006600 0.000682 0.000123 0.000643 0.000118 0.001209 492 1462 5 9
38 15718 -162 -147 -134.0 -13 -335 -142 -298 0 0.0802348 0.2896282 0.6046967 -0.0082706 8.7 8.6 8.5 6.9 8.6 6.8 10.6 0.000672 0.0006670 0.000659 0.000381 0.000658 0.000374 0.000980 9.4 9.3 9.3 7.1 9.2 7.0 11.8 0.000659 0.0006570 0.000682 0.000106 0.000640 0.000106 0.001227 324 1477 5 8
39 15738 -182 -257 -263.0 -363 -476 -275 -365 -22 0.0606654 0.2289628 0.5342466 -0.3335723 5.9 5.8 5.7 4.3 5.7 4.2 7.6 0.000309 0.0002980 0.000273 0.000008 0.000287 -0.000011 0.000652 6.2 6.2 6.1 4.2 6.1 4.1 8.4 0.000659 0.0006555 0.000664 0.000106 0.000639 0.000099 0.001218 365 1903 4 4
40 15748 -176 -212 -202.0 -114 -453 -213 -333 -4 0.0626223 0.2348337 0.5342466 -0.2339613 7.8 7.7 7.5 5.8 7.6 5.7 9.9 0.000740 0.0007350 0.000735 0.000442 0.000725 0.000433 0.001061 8.3 8.3 8.1 6.0 8.2 5.9 10.9 0.000660 0.0006570 0.000638 0.000108 0.000640 0.000100 0.001221 351 1482 6 6
41 15755 -243 -479 -516.0 -486 -715 -528 -486 -66 0.1350294 0.3972603 0.6947162 -0.4375512 5.6 5.5 5.4 2.9 5.4 2.8 8.5 0.001018 0.0010140 0.001015 0.000460 0.000995 0.000446 0.001590 5.9 5.8 5.5 2.8 5.7 2.7 9.3 0.000659 0.0006560 0.000653 0.000105 0.000640 0.000099 0.001223 486 763 3 3
42 15765 -236 -386 -404.0 -464 -640 -416 -471 -60 0.1272016 0.3600783 0.6497065 -0.3346020 6.5 6.5 6.3 4.2 6.4 4.0 9.2 0.001289 0.0012740 0.001250 0.000821 0.001259 0.000797 0.001803 7.0 6.9 6.7 4.4 6.9 4.3 10.0 0.000917 0.0009070 0.000887 0.000543 0.000895 0.000522 0.001322 471 1419 4 4
43 15781 -362 -153 -141.0 -51 -346 -149 -329 0 0.5459883 0.6555773 0.8062622 1.0583806 8.2 8.2 8.0 6.5 8.1 6.4 10.1 0.001041 0.0010300 0.001016 0.000623 0.001017 0.000608 0.001500 8.9 8.9 8.7 7.0 8.8 6.9 11.0 0.000636 0.0006340 0.000655 0.000099 0.000617 0.000092 0.001177 724 831 3 7
44 15803 -327 -247 -253.0 -308 -453 -263 -462 -33 0.3444227 0.4774951 0.6868885 0.5174008 7.3 7.2 7.0 5.3 7.1 5.2 9.5 0.001014 0.0010080 0.001001 0.000473 0.000990 0.000453 0.001577 7.9 7.8 7.5 5.6 7.7 5.5 10.4 0.000520 0.0005200 0.000511 0.000062 0.000505 0.000059 0.000976 654 381 2 5
45 15816 -182 -215 -200.0 -14 -479 -211 -341 -1 0.0665362 0.2465753 0.5636008 -0.2260548 8.8 8.8 8.7 6.9 8.7 6.9 10.8 0.000357 0.0003460 0.000330 0.000126 0.000338 0.000108 0.000627 9.5 9.5 9.6 7.2 9.4 7.1 11.9 0.000660 0.0006570 0.000659 0.000108 0.000640 0.000102 0.001221 364 1836 2 8
46 16031 -232 -207 -221.0 -329 -346 -228 -353 -29 0.3033268 0.4716243 0.7142857 0.4739505 5.4 5.3 5.1 3.4 5.3 3.3 7.6 0.000949 0.0009450 0.000922 0.000340 0.000925 0.000327 0.001581 5.7 5.7 5.5 3.3 5.6 3.2 8.4 0.000660 0.0006570 0.000637 0.000111 0.000640 0.000102 0.001221 465 463 2 4
47 16039 -250 -122 -127.0 -210 -222 -132 -225 -12 0.5733855 0.6555773 0.7984344 1.1539748 5.3 5.3 5.3 4.1 5.3 4.0 6.7 0.000878 0.0008700 0.000881 0.000456 0.000856 0.000440 0.001327 5.6 5.6 5.5 4.2 5.6 4.2 7.2 0.000515 0.0005140 0.000500 0.000110 0.000501 0.000109 0.000922 499 1806 6 4
48 16042 -176 -350 -362.0 -313 -621 -375 -352 -30 0.0841487 0.2837573 0.5968689 -0.4854181 6.6 6.5 6.2 4.2 6.4 4.1 9.3 0.000959 0.0009520 0.000930 0.000390 0.000934 0.000381 0.001554 7.1 7.0 6.8 4.2 6.9 4.0 10.2 0.000659 0.0006550 0.000632 0.000105 0.000638 0.000103 0.001221 352 393 2 5
49 16044 -114 -306 -331.0 -227 -474 -341 -227 -18 0.0802348 0.2935421 0.6105675 -0.4935994 4.8 4.7 4.5 2.9 4.6 2.8 6.9 0.001003 0.0010000 0.001000 0.000384 0.000980 0.000372 0.001635 5.1 5.0 4.7 2.9 4.9 2.7 7.5 0.000647 0.0006430 0.000648 0.000152 0.000628 0.000150 0.001157 227 390 2 2
50 16055 -318 -244 -227.0 -56 -551 -239 -524 0 0.1761252 0.3737769 0.6418787 0.1977097 7.8 7.7 7.6 6.0 7.6 5.9 9.9 0.000701 0.0006880 0.000646 0.000370 0.000677 0.000347 0.001073 8.4 8.3 8.1 6.4 8.2 6.2 10.8 0.000632 0.0006290 0.000609 0.000115 0.000613 0.000110 0.001164 636 1482 3 6
51 16072 -240 -294 -317.0 -455 -461 -326 -471 -56 0.1369863 0.3855186 0.6810176 0.1901266 5.4 5.3 5.3 3.3 5.3 3.2 7.8 0.001063 0.0010530 0.001020 0.000510 0.001034 0.000491 0.001649 5.7 5.7 5.4 3.2 5.6 3.1 8.6 0.000660 0.0006560 0.000639 0.000108 0.000639 0.000105 0.001225 480 1082 3 3
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129 26082 -126 -302 -330.0 -252 -453 -339 -252 -23 0.0919765 0.3033268 0.6379648 -0.4979994 3.5 3.4 3.4 1.9 3.4 1.8 5.2 0.001060 0.0010580 0.001072 0.000390 0.001037 0.000388 0.001743 3.6 3.6 3.4 2.0 3.5 1.9 5.5 0.000644 0.0006420 0.000660 0.000109 0.000625 0.000109 0.001193 252 162 2 1
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146 36813 -248 -188 -193.0 -304 -344 -201 -348 -19 0.3365949 0.4618395 0.6810176 0.5099327 6.4 6.4 6.3 4.6 6.3 4.5 8.5 0.000761 0.0007600 0.000779 0.000165 0.000741 0.000161 0.001369 6.9 6.8 6.6 4.6 6.7 4.5 9.4 0.000659 0.0006550 0.000622 0.000110 0.000638 0.000104 0.001225 496 374 2 5
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148 37471 -79 -160 -149.0 -27 -349 -157 -154 -6 0.0606654 0.2191781 0.5225049 -0.4805940 7.3 7.2 7.1 5.5 7.2 5.4 9.3 0.000997 0.0009880 0.000983 0.000434 0.000970 0.000420 0.001593 7.7 7.7 7.5 5.6 7.6 5.5 10.1 0.000783 0.0007770 0.000743 0.000309 0.000761 0.000301 0.001281 158 353 2 7
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154 37572 -249 -68 -69.0 -114 -125 -73 -126 -5 0.7573386 0.8023483 0.8825832 1.9708789 5.4 5.4 5.3 4.1 5.3 4.1 6.6 0.000901 0.0008880 0.000855 0.000539 0.000876 0.000520 0.001308 5.7 5.7 5.6 4.3 5.6 4.2 7.2 0.000660 0.0006540 0.000629 0.000208 0.000639 0.000198 0.001137 498 2225 7 4
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